| L(s) = 1 | − 2.36·2-s − 3-s + 3.61·4-s − 2.39·5-s + 2.36·6-s − 0.116·7-s − 3.82·8-s + 9-s
+ 5.68·10-s + 6.13·11-s − 3.61·12-s + 13-s + 0.275·14-s + 2.39·15-s + 1.83·16-s + 2.09·17-s
− 2.36·18-s − 7.41·19-s − 8.67·20-s + 0.116·21-s − 14.5·22-s − 1.40·23-s + 3.82·24-s + 0.759·25-s
− 2.36·26-s − 27-s − 0.420·28-s + ⋯
|
| L(s) = 1 | − 1.67·2-s − 0.577·3-s + 1.80·4-s − 1.07·5-s + 0.967·6-s − 0.0439·7-s − 1.35·8-s + 0.333·9-s
+ 1.79·10-s + 1.84·11-s − 1.04·12-s + 0.277·13-s + 0.0736·14-s + 0.619·15-s + 0.459·16-s + 0.508·17-s
− 0.558·18-s − 1.70·19-s − 1.94·20-s + 0.0253·21-s − 3.09·22-s − 0.293·23-s + 0.781·24-s + 0.151·25-s
− 0.464·26-s − 0.192·27-s − 0.0794·28-s + ⋯
|
\[\begin{aligned}
\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr
=\mathstrut & \, \Lambda(2-s)
\end{aligned}
\]
\[\begin{aligned}
\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr
=\mathstrut & \, \Lambda(1-s)
\end{aligned}
\]
\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]
where, for $p \notin \{3,\;13,\;103\}$,
\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;13,\;103\}$, then $F_p$ is a polynomial of degree at most 1.
| $p$ | $F_p$ |
|---|
| bad | 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 + 2.36T + 2T^{2} \) |
| 5 | \( 1 + 2.39T + 5T^{2} \) |
| 7 | \( 1 + 0.116T + 7T^{2} \) |
| 11 | \( 1 - 6.13T + 11T^{2} \) |
| 17 | \( 1 - 2.09T + 17T^{2} \) |
| 19 | \( 1 + 7.41T + 19T^{2} \) |
| 23 | \( 1 + 1.40T + 23T^{2} \) |
| 29 | \( 1 - 7.01T + 29T^{2} \) |
| 31 | \( 1 - 0.944T + 31T^{2} \) |
| 37 | \( 1 - 6.01T + 37T^{2} \) |
| 41 | \( 1 - 7.72T + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 - 2.14T + 53T^{2} \) |
| 59 | \( 1 - 5.08T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 - 8.27T + 67T^{2} \) |
| 71 | \( 1 + 13.4T + 71T^{2} \) |
| 73 | \( 1 + 1.99T + 73T^{2} \) |
| 79 | \( 1 - 1.94T + 79T^{2} \) |
| 83 | \( 1 + 3.21T + 83T^{2} \) |
| 89 | \( 1 + 6.25T + 89T^{2} \) |
| 97 | \( 1 + 6.94T + 97T^{2} \) |
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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
−8.505677890481735231787416298506, −7.84418341764156081950358273698, −7.17665103975292419496229077950, −6.44110254295602782453146061045, −5.99648386527086009937836464783, −4.26351626770972175729188501037, −4.10909617971784025093709134909, −2.63222561805109971302403831871, −1.39923099691693052571183832432, −0.62314679281945546705286803413,
0.62314679281945546705286803413, 1.39923099691693052571183832432, 2.63222561805109971302403831871, 4.10909617971784025093709134909, 4.26351626770972175729188501037, 5.99648386527086009937836464783, 6.44110254295602782453146061045, 7.17665103975292419496229077950, 7.84418341764156081950358273698, 8.505677890481735231787416298506
