L(s) = 1 | + 2-s − 1.98·3-s + 4-s − 5-s − 1.98·6-s − 1.31·7-s + 8-s + 0.924·9-s − 10-s − 3.29·11-s − 1.98·12-s − 2.77·13-s − 1.31·14-s + 1.98·15-s + 16-s + 5.53·17-s + 0.924·18-s + 8.26·19-s − 20-s + 2.60·21-s − 3.29·22-s − 3.94·23-s − 1.98·24-s + 25-s − 2.77·26-s + 4.11·27-s − 1.31·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.14·3-s + 0.5·4-s − 0.447·5-s − 0.808·6-s − 0.496·7-s + 0.353·8-s + 0.308·9-s − 0.316·10-s − 0.994·11-s − 0.571·12-s − 0.770·13-s − 0.351·14-s + 0.511·15-s + 0.250·16-s + 1.34·17-s + 0.217·18-s + 1.89·19-s − 0.223·20-s + 0.567·21-s − 0.703·22-s − 0.821·23-s − 0.404·24-s + 0.200·25-s − 0.544·26-s + 0.791·27-s − 0.248·28-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 | \( 1 + 1.98T + 3T^{2} \) |
| 7 | \( 1 + 1.31T + 7T^{2} \) |
| 11 | \( 1 + 3.29T + 11T^{2} \) |
| 13 | \( 1 + 2.77T + 13T^{2} \) |
| 17 | \( 1 - 5.53T + 17T^{2} \) |
| 19 | \( 1 - 8.26T + 19T^{2} \) |
| 23 | \( 1 + 3.94T + 23T^{2} \) |
| 29 | \( 1 + 10.2T + 29T^{2} \) |
| 31 | \( 1 - 8.12T + 31T^{2} \) |
| 37 | \( 1 - 3.13T + 37T^{2} \) |
| 41 | \( 1 - 6.16T + 41T^{2} \) |
| 43 | \( 1 - 7.50T + 43T^{2} \) |
| 47 | \( 1 + 2.19T + 47T^{2} \) |
| 53 | \( 1 - 1.42T + 53T^{2} \) |
| 59 | \( 1 + 13.2T + 59T^{2} \) |
| 61 | \( 1 - 7.06T + 61T^{2} \) |
| 67 | \( 1 + 7.92T + 67T^{2} \) |
| 71 | \( 1 - 1.57T + 71T^{2} \) |
| 73 | \( 1 - 7.48T + 73T^{2} \) |
| 79 | \( 1 + 8.41T + 79T^{2} \) |
| 83 | \( 1 + 1.48T + 83T^{2} \) |
| 89 | \( 1 + 3.59T + 89T^{2} \) |
| 97 | \( 1 + 4.31T + 97T^{2} \) |
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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
−7.73135272687699467059274979995, −6.94000023769205070258900646720, −5.97328113551352570519212978439, −5.57117628340429889744850119353, −5.05141300783274971985634576760, −4.19677076915469585698194412668, −3.22519328273998456007970136349, −2.64525759496355147520733593151, −1.13064560815708644289777482735, 0,
1.13064560815708644289777482735, 2.64525759496355147520733593151, 3.22519328273998456007970136349, 4.19677076915469585698194412668, 5.05141300783274971985634576760, 5.57117628340429889744850119353, 5.97328113551352570519212978439, 6.94000023769205070258900646720, 7.73135272687699467059274979995