| L(s) = 1 | + 2-s − 0.706·3-s + 4-s + 5-s − 0.706·6-s − 2.05·7-s + 8-s − 2.50·9-s + 10-s + 5.00·11-s − 0.706·12-s + 1.44·13-s − 2.05·14-s − 0.706·15-s + 16-s − 5.11·17-s − 2.50·18-s − 0.185·19-s + 20-s + 1.45·21-s + 5.00·22-s − 2.44·23-s − 0.706·24-s + 25-s + 1.44·26-s + 3.88·27-s − 2.05·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.408·3-s + 0.5·4-s + 0.447·5-s − 0.288·6-s − 0.776·7-s + 0.353·8-s − 0.833·9-s + 0.316·10-s + 1.50·11-s − 0.204·12-s + 0.400·13-s − 0.548·14-s − 0.182·15-s + 0.250·16-s − 1.23·17-s − 0.589·18-s − 0.0424·19-s + 0.223·20-s + 0.316·21-s + 1.06·22-s − 0.510·23-s − 0.144·24-s + 0.200·25-s + 0.283·26-s + 0.748·27-s − 0.388·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 601 | \( 1 + T \) |
| good | 3 | \( 1 + 0.706T + 3T^{2} \) |
| 7 | \( 1 + 2.05T + 7T^{2} \) |
| 11 | \( 1 - 5.00T + 11T^{2} \) |
| 13 | \( 1 - 1.44T + 13T^{2} \) |
| 17 | \( 1 + 5.11T + 17T^{2} \) |
| 19 | \( 1 + 0.185T + 19T^{2} \) |
| 23 | \( 1 + 2.44T + 23T^{2} \) |
| 29 | \( 1 + 2.29T + 29T^{2} \) |
| 31 | \( 1 + 9.99T + 31T^{2} \) |
| 37 | \( 1 + 5.58T + 37T^{2} \) |
| 41 | \( 1 - 2.67T + 41T^{2} \) |
| 43 | \( 1 - 2.86T + 43T^{2} \) |
| 47 | \( 1 + 1.49T + 47T^{2} \) |
| 53 | \( 1 - 9.86T + 53T^{2} \) |
| 59 | \( 1 + 9.79T + 59T^{2} \) |
| 61 | \( 1 - 1.70T + 61T^{2} \) |
| 67 | \( 1 + 4.31T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 + 1.68T + 73T^{2} \) |
| 79 | \( 1 - 3.43T + 79T^{2} \) |
| 83 | \( 1 - 0.478T + 83T^{2} \) |
| 89 | \( 1 + 7.20T + 89T^{2} \) |
| 97 | \( 1 + 4.23T + 97T^{2} \) |
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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
−7.39752103712218294510726889245, −6.75904749888775689243314093667, −6.12134533560762655205904299130, −5.82260013715011014113253056548, −4.86683994903485833201975284267, −3.96461290223578799871802507222, −3.42496660568190813218035762307, −2.40389098770670513202946977181, −1.49374951023143691191529119945, 0,
1.49374951023143691191529119945, 2.40389098770670513202946977181, 3.42496660568190813218035762307, 3.96461290223578799871802507222, 4.86683994903485833201975284267, 5.82260013715011014113253056548, 6.12134533560762655205904299130, 6.75904749888775689243314093667, 7.39752103712218294510726889245
