| L(s) = 1 | − 4·7-s + 2·13-s + 8·19-s − 5·25-s − 4·31-s − 10·37-s + 8·43-s + 9·49-s
+ 14·61-s − 16·67-s − 10·73-s − 4·79-s − 8·91-s + 14·97-s + 101-s + 103-s
+ 107-s + 109-s + 113-s + ⋯
|
| L(s) = 1 | − 1.51·7-s + 0.554·13-s + 1.83·19-s − 25-s − 0.718·31-s − 1.64·37-s + 1.21·43-s + 9/7·49-s
+ 1.79·61-s − 1.95·67-s − 1.17·73-s − 0.450·79-s − 0.838·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s
+ 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯
|
\[\begin{aligned}
\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr
=\mathstrut & \, \Lambda(2-s)
\end{aligned}
\]
\[\begin{aligned}
\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr
=\mathstrut & \, \Lambda(1-s)
\end{aligned}
\]
\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]
where, for $p \notin \{2,\;3\}$,
\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3\}$, then $F_p$ is a polynomial of degree at most 1.
| $p$ | $F_p$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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\[\begin{aligned}
L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}
\end{aligned}\]
Imaginary part of the first few zeros on the critical line
−19.74858543777127, −18.85850998760574, −17.68732582410034, −16.25038600345249, −15.69696813163501, −13.99634105119279, −13.01055982622440, −11.77437667375268, −10.17441103098667, −9.113424945499137, −7.266467310821319, −5.802689552546196, −3.443343367909477,
3.443343367909477, 5.802689552546196, 7.266467310821319, 9.113424945499137, 10.17441103098667, 11.77437667375268, 13.01055982622440, 13.99634105119279, 15.69696813163501, 16.25038600345249, 17.68732582410034, 18.85850998760574, 19.74858543777127
