| L(s) = 1 | − 1.83·2-s − 3.26·3-s + 1.34·4-s + 5.96·6-s − 2.06·7-s + 1.19·8-s + 7.63·9-s − 6.01·11-s − 4.40·12-s + 0.189·13-s + 3.77·14-s − 4.87·16-s − 2.84·17-s − 13.9·18-s + 6.73·21-s + 11.0·22-s − 5.37·23-s − 3.88·24-s − 0.346·26-s − 15.1·27-s − 2.78·28-s + 6.08·29-s + 5.33·31-s + 6.54·32-s + 19.6·33-s + 5.21·34-s + 10.3·36-s + ⋯ |
| L(s) = 1 | − 1.29·2-s − 1.88·3-s + 0.674·4-s + 2.43·6-s − 0.780·7-s + 0.420·8-s + 2.54·9-s − 1.81·11-s − 1.27·12-s + 0.0525·13-s + 1.01·14-s − 1.21·16-s − 0.690·17-s − 3.29·18-s + 1.46·21-s + 2.34·22-s − 1.12·23-s − 0.792·24-s − 0.0679·26-s − 2.91·27-s − 0.526·28-s + 1.12·29-s + 0.958·31-s + 1.15·32-s + 3.41·33-s + 0.893·34-s + 1.71·36-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 | \( 1 + 1.83T + 2T^{2} \) |
| 3 | \( 1 + 3.26T + 3T^{2} \) |
| 7 | \( 1 + 2.06T + 7T^{2} \) |
| 11 | \( 1 + 6.01T + 11T^{2} \) |
| 13 | \( 1 - 0.189T + 13T^{2} \) |
| 17 | \( 1 + 2.84T + 17T^{2} \) |
| 23 | \( 1 + 5.37T + 23T^{2} \) |
| 29 | \( 1 - 6.08T + 29T^{2} \) |
| 31 | \( 1 - 5.33T + 31T^{2} \) |
| 37 | \( 1 + 8.39T + 37T^{2} \) |
| 41 | \( 1 - 7.13T + 41T^{2} \) |
| 43 | \( 1 + 3.05T + 43T^{2} \) |
| 47 | \( 1 - 0.574T + 47T^{2} \) |
| 53 | \( 1 + 3.40T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 + 4.01T + 61T^{2} \) |
| 67 | \( 1 - 3.21T + 67T^{2} \) |
| 71 | \( 1 + 15.6T + 71T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 + 3.19T + 79T^{2} \) |
| 83 | \( 1 + 6.11T + 83T^{2} \) |
| 89 | \( 1 + 15.6T + 89T^{2} \) |
| 97 | \( 1 - 1.68T + 97T^{2} \) |
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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
−7.35724831887360562582659665393, −6.73188018284690713974050194625, −6.19328616384278778905428596736, −5.43042231851940701263394742496, −4.79670088765407977548057360545, −4.14188980777488059804117436998, −2.75964618251841336339160465736, −1.76504636284297561863589082479, −0.62411944757275302275721076084, 0,
0.62411944757275302275721076084, 1.76504636284297561863589082479, 2.75964618251841336339160465736, 4.14188980777488059804117436998, 4.79670088765407977548057360545, 5.43042231851940701263394742496, 6.19328616384278778905428596736, 6.73188018284690713974050194625, 7.35724831887360562582659665393
