| L(s) = 1 | + 2.13·2-s + 3.13·3-s + 2.55·4-s + 6.69·6-s − 1.41·7-s + 1.18·8-s + 6.84·9-s − 1.68·11-s + 8.01·12-s + 4.54·13-s − 3.02·14-s − 2.58·16-s + 3.25·17-s + 14.6·18-s − 0.0290·19-s − 4.44·21-s − 3.60·22-s − 2.28·23-s + 3.71·24-s + 9.69·26-s + 12.0·27-s − 3.61·28-s + 8.95·29-s + 4.17·31-s − 7.87·32-s − 5.30·33-s + 6.95·34-s + ⋯ |
| L(s) = 1 | + 1.50·2-s + 1.81·3-s + 1.27·4-s + 2.73·6-s − 0.535·7-s + 0.418·8-s + 2.28·9-s − 0.509·11-s + 2.31·12-s + 1.25·13-s − 0.808·14-s − 0.645·16-s + 0.790·17-s + 3.44·18-s − 0.00665·19-s − 0.970·21-s − 0.768·22-s − 0.476·23-s + 0.758·24-s + 1.90·26-s + 2.31·27-s − 0.684·28-s + 1.66·29-s + 0.749·31-s − 1.39·32-s − 0.922·33-s + 1.19·34-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
| \(L(1)\) |
\(\approx\) |
\(9.308732253\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.308732253\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.13T + 2T^{2} \) |
| 3 | \( 1 - 3.13T + 3T^{2} \) |
| 7 | \( 1 + 1.41T + 7T^{2} \) |
| 11 | \( 1 + 1.68T + 11T^{2} \) |
| 13 | \( 1 - 4.54T + 13T^{2} \) |
| 17 | \( 1 - 3.25T + 17T^{2} \) |
| 19 | \( 1 + 0.0290T + 19T^{2} \) |
| 23 | \( 1 + 2.28T + 23T^{2} \) |
| 29 | \( 1 - 8.95T + 29T^{2} \) |
| 31 | \( 1 - 4.17T + 31T^{2} \) |
| 37 | \( 1 + 0.842T + 37T^{2} \) |
| 41 | \( 1 - 1.39T + 41T^{2} \) |
| 43 | \( 1 - 9.59T + 43T^{2} \) |
| 47 | \( 1 - 4.29T + 47T^{2} \) |
| 53 | \( 1 + 7.25T + 53T^{2} \) |
| 59 | \( 1 + 1.64T + 59T^{2} \) |
| 61 | \( 1 + 2.40T + 61T^{2} \) |
| 67 | \( 1 + 1.37T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 - 1.36T + 79T^{2} \) |
| 83 | \( 1 + 1.41T + 83T^{2} \) |
| 89 | \( 1 - 16.0T + 89T^{2} \) |
| 97 | \( 1 - 13.8T + 97T^{2} \) |
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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
−7.965944916086077886421267519899, −7.44668060950044157917534162275, −6.42315440968952135118633592556, −6.03821021632819360384326175861, −4.90820386517794746890129088593, −4.20648300193027873271540906251, −3.57069133136260174636774495436, −2.98290343951990892010875049232, −2.48717231197133441019022180486, −1.31179924581610809062780948026,
1.31179924581610809062780948026, 2.48717231197133441019022180486, 2.98290343951990892010875049232, 3.57069133136260174636774495436, 4.20648300193027873271540906251, 4.90820386517794746890129088593, 6.03821021632819360384326175861, 6.42315440968952135118633592556, 7.44668060950044157917534162275, 7.965944916086077886421267519899
