| L(s) = 1 | − 1.69·2-s + 0.918·3-s + 0.871·4-s − 1.55·6-s − 3.12·7-s + 1.91·8-s − 2.15·9-s + 5.68·11-s + 0.799·12-s + 5.35·13-s + 5.29·14-s − 4.98·16-s − 7.27·17-s + 3.65·18-s − 2.87·21-s − 9.63·22-s − 1.74·23-s + 1.75·24-s − 9.07·26-s − 4.73·27-s − 2.72·28-s + 7.78·29-s − 2.11·31-s + 4.61·32-s + 5.22·33-s + 12.3·34-s − 1.87·36-s + ⋯ |
| L(s) = 1 | − 1.19·2-s + 0.530·3-s + 0.435·4-s − 0.635·6-s − 1.18·7-s + 0.676·8-s − 0.718·9-s + 1.71·11-s + 0.230·12-s + 1.48·13-s + 1.41·14-s − 1.24·16-s − 1.76·17-s + 0.861·18-s − 0.626·21-s − 2.05·22-s − 0.364·23-s + 0.358·24-s − 1.77·26-s − 0.911·27-s − 0.514·28-s + 1.44·29-s − 0.380·31-s + 0.816·32-s + 0.909·33-s + 2.11·34-s − 0.313·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8846453465\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8846453465\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 1.69T + 2T^{2} \) |
| 3 | \( 1 - 0.918T + 3T^{2} \) |
| 7 | \( 1 + 3.12T + 7T^{2} \) |
| 11 | \( 1 - 5.68T + 11T^{2} \) |
| 13 | \( 1 - 5.35T + 13T^{2} \) |
| 17 | \( 1 + 7.27T + 17T^{2} \) |
| 23 | \( 1 + 1.74T + 23T^{2} \) |
| 29 | \( 1 - 7.78T + 29T^{2} \) |
| 31 | \( 1 + 2.11T + 31T^{2} \) |
| 37 | \( 1 + 4.72T + 37T^{2} \) |
| 41 | \( 1 - 0.302T + 41T^{2} \) |
| 43 | \( 1 + 3.54T + 43T^{2} \) |
| 47 | \( 1 - 0.288T + 47T^{2} \) |
| 53 | \( 1 + 9.14T + 53T^{2} \) |
| 59 | \( 1 + 0.589T + 59T^{2} \) |
| 61 | \( 1 - 7.80T + 61T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 - 11.7T + 71T^{2} \) |
| 73 | \( 1 - 1.03T + 73T^{2} \) |
| 79 | \( 1 - 5.70T + 79T^{2} \) |
| 83 | \( 1 - 3.71T + 83T^{2} \) |
| 89 | \( 1 + 5.54T + 89T^{2} \) |
| 97 | \( 1 - 6.81T + 97T^{2} \) |
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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
−8.158424278068731558771692796527, −6.97897027654373365308580764157, −6.49355384358339253643189310732, −6.19413830430183404741084987314, −4.87635017753182718321285544804, −3.86947894628380650399645017095, −3.55397779956669095996952383069, −2.42949925413009455964693108523, −1.56251029683393847729750112177, −0.54764952739527275739709289223,
0.54764952739527275739709289223, 1.56251029683393847729750112177, 2.42949925413009455964693108523, 3.55397779956669095996952383069, 3.86947894628380650399645017095, 4.87635017753182718321285544804, 6.19413830430183404741084987314, 6.49355384358339253643189310732, 6.97897027654373365308580764157, 8.158424278068731558771692796527
