| L(s) = 1 | + 2-s + 0.956·3-s + 4-s + 0.956·6-s − 2.15·7-s + 8-s − 2.08·9-s + 1.30·11-s + 0.956·12-s − 2.15·14-s + 16-s − 2.87·17-s − 2.08·18-s − 2.65·19-s − 2.05·21-s + 1.30·22-s + 5.11·23-s + 0.956·24-s − 4.86·27-s − 2.15·28-s + 7.96·29-s − 4.12·31-s + 32-s + 1.24·33-s − 2.87·34-s − 2.08·36-s + 2.98·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.552·3-s + 0.5·4-s + 0.390·6-s − 0.813·7-s + 0.353·8-s − 0.695·9-s + 0.393·11-s + 0.276·12-s − 0.574·14-s + 0.250·16-s − 0.697·17-s − 0.491·18-s − 0.609·19-s − 0.448·21-s + 0.278·22-s + 1.06·23-s + 0.195·24-s − 0.936·27-s − 0.406·28-s + 1.47·29-s − 0.741·31-s + 0.176·32-s + 0.217·33-s − 0.492·34-s − 0.347·36-s + 0.491·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.318589962\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.318589962\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 0.956T + 3T^{2} \) |
| 7 | \( 1 + 2.15T + 7T^{2} \) |
| 11 | \( 1 - 1.30T + 11T^{2} \) |
| 17 | \( 1 + 2.87T + 17T^{2} \) |
| 19 | \( 1 + 2.65T + 19T^{2} \) |
| 23 | \( 1 - 5.11T + 23T^{2} \) |
| 29 | \( 1 - 7.96T + 29T^{2} \) |
| 31 | \( 1 + 4.12T + 31T^{2} \) |
| 37 | \( 1 - 2.98T + 37T^{2} \) |
| 41 | \( 1 - 8.69T + 41T^{2} \) |
| 43 | \( 1 - 3.35T + 43T^{2} \) |
| 47 | \( 1 - 7.30T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 + 14.6T + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 + 11.0T + 67T^{2} \) |
| 71 | \( 1 - 14.0T + 71T^{2} \) |
| 73 | \( 1 + 3.83T + 73T^{2} \) |
| 79 | \( 1 + 8.94T + 79T^{2} \) |
| 83 | \( 1 - 13.9T + 83T^{2} \) |
| 89 | \( 1 - 15.6T + 89T^{2} \) |
| 97 | \( 1 - 5.05T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.66583638811217237976868279781, −7.00955778557870299842054964828, −6.28844928851765446707344387373, −5.86921625413232632954151397445, −4.89091003134231618678886715723, −4.18742592056494692678593992685, −3.43737759158710336225807720783, −2.76093150666703162955871898913, −2.16636593548677153321948236334, −0.75436807116018258415843557168,
0.75436807116018258415843557168, 2.16636593548677153321948236334, 2.76093150666703162955871898913, 3.43737759158710336225807720783, 4.18742592056494692678593992685, 4.89091003134231618678886715723, 5.86921625413232632954151397445, 6.28844928851765446707344387373, 7.00955778557870299842054964828, 7.66583638811217237976868279781
