| L(s) = 1 | + 2-s + 0.460·3-s + 4-s + 0.645·5-s + 0.460·6-s + 1.69·7-s + 8-s − 2.78·9-s
+ 0.645·10-s + 0.878·11-s + 0.460·12-s − 2.21·13-s + 1.69·14-s + 0.297·15-s + 16-s − 3.24·17-s
− 2.78·18-s + 19-s + 0.645·20-s + 0.782·21-s + 0.878·22-s − 0.250·23-s + 0.460·24-s − 4.58·25-s
− 2.21·26-s − 2.66·27-s + 1.69·28-s + ⋯
|
| L(s) = 1 | + 0.707·2-s + 0.266·3-s + 0.5·4-s + 0.288·5-s + 0.188·6-s + 0.641·7-s + 0.353·8-s − 0.929·9-s
+ 0.204·10-s + 0.264·11-s + 0.133·12-s − 0.613·13-s + 0.453·14-s + 0.0767·15-s + 0.250·16-s − 0.787·17-s
− 0.657·18-s + 0.229·19-s + 0.144·20-s + 0.170·21-s + 0.187·22-s − 0.0522·23-s + 0.0940·24-s − 0.916·25-s
− 0.433·26-s − 0.513·27-s + 0.320·28-s + ⋯
|
\[\begin{aligned}
\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr
=\mathstrut & -\, \Lambda(2-s)
\end{aligned}
\]
\[\begin{aligned}
\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr
=\mathstrut & -\, \Lambda(1-s)
\end{aligned}
\]
\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]
where, for $p \notin \{2,\;19,\;211\}$,
\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;19,\;211\}$, then $F_p$ is a polynomial of degree at most 1.
| $p$ | $F_p$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 - 0.460T + 3T^{2} \) |
| 5 | \( 1 - 0.645T + 5T^{2} \) |
| 7 | \( 1 - 1.69T + 7T^{2} \) |
| 11 | \( 1 - 0.878T + 11T^{2} \) |
| 13 | \( 1 + 2.21T + 13T^{2} \) |
| 17 | \( 1 + 3.24T + 17T^{2} \) |
| 23 | \( 1 + 0.250T + 23T^{2} \) |
| 29 | \( 1 + 6.95T + 29T^{2} \) |
| 31 | \( 1 + 5.17T + 31T^{2} \) |
| 37 | \( 1 - 4.12T + 37T^{2} \) |
| 41 | \( 1 + 4.51T + 41T^{2} \) |
| 43 | \( 1 - 0.863T + 43T^{2} \) |
| 47 | \( 1 + 8.45T + 47T^{2} \) |
| 53 | \( 1 - 1.10T + 53T^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 - 7.83T + 61T^{2} \) |
| 67 | \( 1 + 4.12T + 67T^{2} \) |
| 71 | \( 1 + 6.15T + 71T^{2} \) |
| 73 | \( 1 + 0.0206T + 73T^{2} \) |
| 79 | \( 1 - 3.73T + 79T^{2} \) |
| 83 | \( 1 - 12.2T + 83T^{2} \) |
| 89 | \( 1 - 0.944T + 89T^{2} \) |
| 97 | \( 1 + 8.08T + 97T^{2} \) |
| show more | |
| show less | |
\[\begin{aligned}
L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}
\end{aligned}\]
Imaginary part of the first few zeros on the critical line
−7.56886674093581785274509066700, −6.67600075612039521709149611847, −5.98076308140867186976706231710, −5.35370989681612695296676369319, −4.73939562608908792029248037810, −3.89161979351856341261839978238, −3.16132366299433694306930238454, −2.23967873457407165815545158129, −1.68366339972686948143125319175, 0,
1.68366339972686948143125319175, 2.23967873457407165815545158129, 3.16132366299433694306930238454, 3.89161979351856341261839978238, 4.73939562608908792029248037810, 5.35370989681612695296676369319, 5.98076308140867186976706231710, 6.67600075612039521709149611847, 7.56886674093581785274509066700
