| L(s) = 1 | − 2.35·2-s + 2.46·3-s + 3.56·4-s − 5.80·6-s − 4.19·7-s − 3.67·8-s + 3.06·9-s − 0.00154·11-s + 8.76·12-s + 3.62·13-s + 9.88·14-s + 1.55·16-s + 1.85·17-s − 7.22·18-s − 0.186·19-s − 10.3·21-s + 0.00363·22-s + 6.05·23-s − 9.06·24-s − 8.53·26-s + 0.162·27-s − 14.9·28-s − 10.2·29-s − 1.07·31-s + 3.69·32-s − 0.00379·33-s − 4.36·34-s + ⋯ |
| L(s) = 1 | − 1.66·2-s + 1.42·3-s + 1.78·4-s − 2.37·6-s − 1.58·7-s − 1.30·8-s + 1.02·9-s − 0.000464·11-s + 2.53·12-s + 1.00·13-s + 2.64·14-s + 0.388·16-s + 0.448·17-s − 1.70·18-s − 0.0428·19-s − 2.25·21-s + 0.000775·22-s + 1.26·23-s − 1.84·24-s − 1.67·26-s + 0.0313·27-s − 2.82·28-s − 1.90·29-s − 0.192·31-s + 0.652·32-s − 0.000661·33-s − 0.748·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.157397239\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.157397239\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 + 2.35T + 2T^{2} \) |
| 3 | \( 1 - 2.46T + 3T^{2} \) |
| 7 | \( 1 + 4.19T + 7T^{2} \) |
| 11 | \( 1 + 0.00154T + 11T^{2} \) |
| 13 | \( 1 - 3.62T + 13T^{2} \) |
| 17 | \( 1 - 1.85T + 17T^{2} \) |
| 19 | \( 1 + 0.186T + 19T^{2} \) |
| 23 | \( 1 - 6.05T + 23T^{2} \) |
| 29 | \( 1 + 10.2T + 29T^{2} \) |
| 31 | \( 1 + 1.07T + 31T^{2} \) |
| 37 | \( 1 - 6.88T + 37T^{2} \) |
| 41 | \( 1 + 7.15T + 41T^{2} \) |
| 43 | \( 1 + 7.53T + 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 + 6.06T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 - 6.10T + 61T^{2} \) |
| 67 | \( 1 - 8.53T + 67T^{2} \) |
| 71 | \( 1 + 11.9T + 71T^{2} \) |
| 73 | \( 1 - 7.27T + 73T^{2} \) |
| 79 | \( 1 - 14.7T + 79T^{2} \) |
| 83 | \( 1 - 1.15T + 83T^{2} \) |
| 89 | \( 1 - 7.45T + 89T^{2} \) |
| 97 | \( 1 + 8.29T + 97T^{2} \) |
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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
−8.271134767315923233115899807672, −7.61802867047274103060826904595, −6.97813479039474651181902544604, −6.41915898119269588543663940531, −5.47422520462829046352209739771, −3.87875699968297377295988702289, −3.36475915205160032689969078890, −2.63979005319578724934375466140, −1.77380787419541749721658829742, −0.67184473788071977610897952208,
0.67184473788071977610897952208, 1.77380787419541749721658829742, 2.63979005319578724934375466140, 3.36475915205160032689969078890, 3.87875699968297377295988702289, 5.47422520462829046352209739771, 6.41915898119269588543663940531, 6.97813479039474651181902544604, 7.61802867047274103060826904595, 8.271134767315923233115899807672
