| L(s) = 1 | − 2.61·2-s + 2.23·3-s + 4.85·4-s − 5.85·6-s − 4.23·7-s − 7.47·8-s + 2.00·9-s + 5.47·11-s + 10.8·12-s + 2·13-s + 11.0·14-s + 9.85·16-s − 4.85·17-s − 5.23·18-s − 9.47·21-s − 14.3·22-s − 1.23·23-s − 16.7·24-s − 5.23·26-s − 2.23·27-s − 20.5·28-s − 6·29-s + 1.85·31-s − 10.8·32-s + 12.2·33-s + 12.7·34-s + 9.70·36-s + ⋯ |
| L(s) = 1 | − 1.85·2-s + 1.29·3-s + 2.42·4-s − 2.38·6-s − 1.60·7-s − 2.64·8-s + 0.666·9-s + 1.64·11-s + 3.13·12-s + 0.554·13-s + 2.96·14-s + 2.46·16-s − 1.17·17-s − 1.23·18-s − 2.06·21-s − 3.05·22-s − 0.257·23-s − 3.41·24-s − 1.02·26-s − 0.430·27-s − 3.88·28-s − 1.11·29-s + 0.333·31-s − 1.91·32-s + 2.13·33-s + 2.17·34-s + 1.61·36-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.61T + 2T^{2} \) |
| 3 | \( 1 - 2.23T + 3T^{2} \) |
| 7 | \( 1 + 4.23T + 7T^{2} \) |
| 11 | \( 1 - 5.47T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 4.85T + 17T^{2} \) |
| 23 | \( 1 + 1.23T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 1.85T + 31T^{2} \) |
| 37 | \( 1 + 4.14T + 37T^{2} \) |
| 41 | \( 1 - 6.38T + 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 + 7.61T + 47T^{2} \) |
| 53 | \( 1 - 6.38T + 53T^{2} \) |
| 59 | \( 1 - 2.76T + 59T^{2} \) |
| 61 | \( 1 + 5.56T + 61T^{2} \) |
| 67 | \( 1 - 8.70T + 67T^{2} \) |
| 71 | \( 1 - 4.52T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 + T + 79T^{2} \) |
| 83 | \( 1 - 1.09T + 83T^{2} \) |
| 89 | \( 1 + 3.70T + 89T^{2} \) |
| 97 | \( 1 + 7.38T + 97T^{2} \) |
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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
−7.49194715309917211490484652372, −6.99773052779431686257558485758, −6.38023084479049233426945192922, −5.93225916557088466091950702047, −4.05430363955048767070631281066, −3.58810936415103998876882393207, −2.74520327274596273835446013246, −2.10145415396515924583231871878, −1.17378629497462513275111252825, 0,
1.17378629497462513275111252825, 2.10145415396515924583231871878, 2.74520327274596273835446013246, 3.58810936415103998876882393207, 4.05430363955048767070631281066, 5.93225916557088466091950702047, 6.38023084479049233426945192922, 6.99773052779431686257558485758, 7.49194715309917211490484652372
