| L(s) = 1 | − 1.55·2-s + 3-s + 0.403·4-s − 3.10·5-s − 1.55·6-s − 2.24·7-s + 2.47·8-s + 9-s + 4.81·10-s − 3.34·11-s + 0.403·12-s − 2.20·13-s + 3.47·14-s − 3.10·15-s − 4.64·16-s + 17-s − 1.55·18-s + 2.95·19-s − 1.25·20-s − 2.24·21-s + 5.19·22-s + 5.25·23-s + 2.47·24-s + 4.65·25-s + 3.41·26-s + 27-s − 0.905·28-s + ⋯ |
| L(s) = 1 | − 1.09·2-s + 0.577·3-s + 0.201·4-s − 1.38·5-s − 0.632·6-s − 0.848·7-s + 0.875·8-s + 0.333·9-s + 1.52·10-s − 1.00·11-s + 0.116·12-s − 0.611·13-s + 0.930·14-s − 0.802·15-s − 1.16·16-s + 0.242·17-s − 0.365·18-s + 0.677·19-s − 0.280·20-s − 0.489·21-s + 1.10·22-s + 1.09·23-s + 0.505·24-s + 0.930·25-s + 0.669·26-s + 0.192·27-s − 0.171·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 + T \) |
| good | 2 | \( 1 + 1.55T + 2T^{2} \) |
| 5 | \( 1 + 3.10T + 5T^{2} \) |
| 7 | \( 1 + 2.24T + 7T^{2} \) |
| 11 | \( 1 + 3.34T + 11T^{2} \) |
| 13 | \( 1 + 2.20T + 13T^{2} \) |
| 19 | \( 1 - 2.95T + 19T^{2} \) |
| 23 | \( 1 - 5.25T + 23T^{2} \) |
| 29 | \( 1 + 5.08T + 29T^{2} \) |
| 31 | \( 1 - 4.47T + 31T^{2} \) |
| 37 | \( 1 - 3.83T + 37T^{2} \) |
| 41 | \( 1 + 10.2T + 41T^{2} \) |
| 43 | \( 1 + 1.83T + 43T^{2} \) |
| 47 | \( 1 + 4.85T + 47T^{2} \) |
| 53 | \( 1 - 9.63T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 - 0.654T + 61T^{2} \) |
| 67 | \( 1 - 7.76T + 67T^{2} \) |
| 71 | \( 1 - 8.94T + 71T^{2} \) |
| 73 | \( 1 + 3.07T + 73T^{2} \) |
| 79 | \( 1 - 0.816T + 79T^{2} \) |
| 83 | \( 1 - 16.5T + 83T^{2} \) |
| 89 | \( 1 - 8.72T + 89T^{2} \) |
| 97 | \( 1 - 15.2T + 97T^{2} \) |
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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
−7.67859995639354778430404284695, −7.24004776164062748883622762237, −6.50298295965270796124483491059, −5.14726484945515388318682206692, −4.69715959076995576666024137965, −3.62545569902507567052427196882, −3.17935593324406364454212598867, −2.18829659526161078982058085715, −0.851603147311639423665664205888, 0,
0.851603147311639423665664205888, 2.18829659526161078982058085715, 3.17935593324406364454212598867, 3.62545569902507567052427196882, 4.69715959076995576666024137965, 5.14726484945515388318682206692, 6.50298295965270796124483491059, 7.24004776164062748883622762237, 7.67859995639354778430404284695
