| L(s) = 1 | − 2.49·2-s − 1.22·3-s + 4.20·4-s + 3.04·6-s − 0.136·7-s − 5.48·8-s − 1.50·9-s − 0.905·11-s
− 5.13·12-s + 0.123·13-s + 0.339·14-s + 5.26·16-s − 1.26·17-s + 3.75·18-s − 2.13·19-s + 0.166·21-s
+ 2.25·22-s − 6.64·23-s + 6.70·24-s − 0.308·26-s + 5.50·27-s − 0.572·28-s + 5.36·29-s − 9.78·31-s
− 2.13·32-s + 1.10·33-s + 3.13·34-s + ⋯
|
| L(s) = 1 | − 1.76·2-s − 0.705·3-s + 2.10·4-s + 1.24·6-s − 0.0514·7-s − 1.94·8-s − 0.502·9-s − 0.272·11-s
− 1.48·12-s + 0.0343·13-s + 0.0906·14-s + 1.31·16-s − 0.305·17-s + 0.884·18-s − 0.489·19-s + 0.0363·21-s
+ 0.480·22-s − 1.38·23-s + 1.36·24-s − 0.0604·26-s + 1.05·27-s − 0.108·28-s + 0.996·29-s − 1.75·31-s
− 0.377·32-s + 0.192·33-s + 0.538·34-s + ⋯
|
\[\begin{aligned}
\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr
=\mathstrut & \, \Lambda(2-s)
\end{aligned}
\]
\[\begin{aligned}
\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr
=\mathstrut & \, \Lambda(1-s)
\end{aligned}
\]
\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]
where, for $p \notin \{5,\;241\}$,
\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{5,\;241\}$, then $F_p$ is a polynomial of degree at most 1.
| $p$ | $F_p$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 + 2.49T + 2T^{2} \) |
| 3 | \( 1 + 1.22T + 3T^{2} \) |
| 7 | \( 1 + 0.136T + 7T^{2} \) |
| 11 | \( 1 + 0.905T + 11T^{2} \) |
| 13 | \( 1 - 0.123T + 13T^{2} \) |
| 17 | \( 1 + 1.26T + 17T^{2} \) |
| 19 | \( 1 + 2.13T + 19T^{2} \) |
| 23 | \( 1 + 6.64T + 23T^{2} \) |
| 29 | \( 1 - 5.36T + 29T^{2} \) |
| 31 | \( 1 + 9.78T + 31T^{2} \) |
| 37 | \( 1 + 5.76T + 37T^{2} \) |
| 41 | \( 1 - 6.43T + 41T^{2} \) |
| 43 | \( 1 - 3.18T + 43T^{2} \) |
| 47 | \( 1 + 12.9T + 47T^{2} \) |
| 53 | \( 1 + 3.90T + 53T^{2} \) |
| 59 | \( 1 - 8.15T + 59T^{2} \) |
| 61 | \( 1 + 14.3T + 61T^{2} \) |
| 67 | \( 1 - 4.89T + 67T^{2} \) |
| 71 | \( 1 - 4.32T + 71T^{2} \) |
| 73 | \( 1 + 5.64T + 73T^{2} \) |
| 79 | \( 1 - 1.43T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 + 13.7T + 89T^{2} \) |
| 97 | \( 1 + 13.6T + 97T^{2} \) |
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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
−8.187498945855526615591571262592, −7.59469657741235213549529837197, −6.71616509761784313671102938160, −6.25491631843808871599599746739, −5.51670478869478182179342198943, −4.56366833790394423537593237656, −3.35327772551603931393493047998, −2.36662257398082114027121702059, −1.56648190417741679576757448773, −0.30394124144063704065703729827,
0.30394124144063704065703729827, 1.56648190417741679576757448773, 2.36662257398082114027121702059, 3.35327772551603931393493047998, 4.56366833790394423537593237656, 5.51670478869478182179342198943, 6.25491631843808871599599746739, 6.71616509761784313671102938160, 7.59469657741235213549529837197, 8.187498945855526615591571262592
