Michael Cole produced an excellent article entitled '"My Name Is" Cipher Motivation', indicating that the Zodiac Killer's third cipher was a direct response to an article by Will Stevens on October 22nd 1969 in which Dr DCB Marsh, president of The American Cryptogram Association, laid down a challenge to Zodiac. In the article Dr Marsh stated "The killer wouldn't dare, as he claimed in letters to the newspapers, to reveal his name in a cipher to established cryptogram experts. "Zodiac" had not done this, Marsh suggested, because to tell the complete truth in relation to his name, in cipher code, would lead to his capture. I invite 'Zodiac' to send The American Cryptogram Association a cipher code". We now know, that six months later, Zodiac did respond with a 13 symbol cipher, that began with the line "This is the Zodiac speaking. By the way have you cracked the last cipher I sent you. My name is....".
To view this article please visit the website Zodiac Revisited http://zodiacrevisited.com/name-cipher-motivation/
Michael Cole also stated something very interesting in the comments section that pricked my interest, notably "I don’t assign much meaning to some version of “Zodiac” showing up in the last line of the 340. Based on the 408, we should expect part of that line to be filler. He was likely just playing around with his persona’s name in the filler".
If we take ourselves back to the 408 cipher, mailed in three parts to the Chronicle, Examiner and Herald on July 31st 1969, Zodiac stated "In this cipher is my idenity". He stated his identity, but not necessarily his name. Four days later, on August 4th 1969, the 'Debut of Zodiac' letter arrived at the San Francisco Examiner revealing his identity in the opening line, stating "This is the Zodiac Speaking". When the 408 cipher was decoded it revealed that Zodiac would not give his name, only his identity. But where was his identity 'Zodiac' in the 408 cipher? Was it somehow enciphered in the undecoded 18 letters at the foot of the 408 cipher? If the final line of the 340 cipher was also filler, as Michael Cole inferred - "Based on the 408, we should expect part of that line to be filler. He was likely just playing around with his persona’s name in the filler" - then it is also possible that the identity name of 'Zodiac' was concealed at the base of the 408 in similar fashion.

I concur, that it is very likely the 13 symbol cipher was the response to the article, but couldn't help thinking that the 340 cipher was somehow a temporary response - a secondary filler if you like. Was the Zodiac Killer already preparing the 340 Cipher and after reading this article dropped in his identity on the bottom line as filler, which he would ultimately connect to his next cipher on April 20th 1970 (The My Name is Cipher). However, for this to be the case, his name in the 13 symbol cipher would also have to be present in the 340 cipher in some form.
The Zodiac Killer responded to the newspapers on a regular basis, so it certainly is most likely he responded to this one also. But would the Zodiac Killer really reveal his name? Michael Cole makes another valid point: "This relationship between the Examiner article and the My Name Is cipher may well mean there is an even higher likelihood that the cryptogram does indeed encipher some form of the killer's name; the reasoning being that the Zodiac would have felt compelled to abide by the rules of the challenge in order to legitimately win the dare and thereby prove his intellectual superiority to Dr. Marsh, the ACA membership, and the rest of the cipher-solving world". The key phrase here is "some form of the killer's name". and the Zodiac Killer certainly loved a challenge. But when he devised the 13 symbol cipher, did he ultimately give us a name, just not the one we were hoping for? Did the Zodiac Killer use something within the actual article itself as a sort of joke, displaying his superiority and toying with Dr Marsh?

In the newspaper article, Dr Marsh stated "There were 18 garbled letters at the end- never deciphered". So, was it likely that the Zodiac would reply in kind, by leaving 18 garbled letters at the end of the 340 cipher, that would ultimately reveal his name. Five months would pass since the 340 cipher with no solution - so just like the 'Little List' letter was mailed to compliment the unsolved 'Button' letter code and Mount Diablo map, did Zodiac mail the 13 symbol cipher to push us in the right direction? Eight symbols from the 340 cipher can be connected to the 13 symbol cipher as follows. The Zodiac Killer was laid a challenge by Dr Marsh to reveal his name, but in the 340 cipher Zodiac seemingly only revealed his identity in the form of ZO (TRIANGLE) AIK. However, Zodiac was the ultimate egotist, so was it possible the clue lay in these six symbols, formed in such a way they would eventually become integrated into his 13 symbol cipher and reveal in part his true name?

http://zodiacrevisited.com/name-cipher-motivation/
Other than an article about the phony Jim Dunbar TV phone call, the previous newspaper article to this Zodiac correspondence, was published by the San Francisco Chronicle on January 28th 1970, entitled 'Yellow Cab Sets reward for Zodiac.' Hence Zodiac's reply, asking how much money was on his head now. The article begins with the actual sum of the reward, but this was in January, and it was now April. In addition, the newspaper article detailed an additional sum that was likely to be offered. The article read "Yellow Cab Company yesterday posted a $1000 reward for the Zodiac Killer. Specifically the money is being offered for "information leading to the arrest and conviction of the person(s) responsible" for the deaths of Yellow Cab drivers Charles Jarman and Paul Stine. The Teamsters Union, which represents Yellow Cab's drivers, is also considering offering a reward in the two cases. The subject will be discussed and voted on at a meeting of drivers set for February 8."