L(s) = 1 | − 2-s + 2.35·3-s + 4-s − 2.63·5-s − 2.35·6-s + 0.368·7-s − 8-s + 2.53·9-s + 2.63·10-s − 4.47·11-s + 2.35·12-s − 3.72·13-s − 0.368·14-s − 6.20·15-s + 16-s − 4.24·17-s − 2.53·18-s + 2.19·19-s − 2.63·20-s + 0.867·21-s + 4.47·22-s − 4.44·23-s − 2.35·24-s + 1.95·25-s + 3.72·26-s − 1.08·27-s + 0.368·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.35·3-s + 0.5·4-s − 1.17·5-s − 0.960·6-s + 0.139·7-s − 0.353·8-s + 0.846·9-s + 0.834·10-s − 1.35·11-s + 0.679·12-s − 1.03·13-s − 0.0984·14-s − 1.60·15-s + 0.250·16-s − 1.02·17-s − 0.598·18-s + 0.504·19-s − 0.589·20-s + 0.189·21-s + 0.955·22-s − 0.926·23-s − 0.480·24-s + 0.391·25-s + 0.731·26-s − 0.208·27-s + 0.0696·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 862 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 862 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 431 | \( 1 + T \) |
good | 3 | \( 1 - 2.35T + 3T^{2} \) |
| 5 | \( 1 + 2.63T + 5T^{2} \) |
| 7 | \( 1 - 0.368T + 7T^{2} \) |
| 11 | \( 1 + 4.47T + 11T^{2} \) |
| 13 | \( 1 + 3.72T + 13T^{2} \) |
| 17 | \( 1 + 4.24T + 17T^{2} \) |
| 19 | \( 1 - 2.19T + 19T^{2} \) |
| 23 | \( 1 + 4.44T + 23T^{2} \) |
| 29 | \( 1 + 0.757T + 29T^{2} \) |
| 31 | \( 1 - 6.05T + 31T^{2} \) |
| 37 | \( 1 - 3.27T + 37T^{2} \) |
| 41 | \( 1 + 2.11T + 41T^{2} \) |
| 43 | \( 1 + 11.5T + 43T^{2} \) |
| 47 | \( 1 - 0.475T + 47T^{2} \) |
| 53 | \( 1 - 5.48T + 53T^{2} \) |
| 59 | \( 1 - 5.59T + 59T^{2} \) |
| 61 | \( 1 - 14.0T + 61T^{2} \) |
| 67 | \( 1 + 5.79T + 67T^{2} \) |
| 71 | \( 1 + 0.983T + 71T^{2} \) |
| 73 | \( 1 + 2.08T + 73T^{2} \) |
| 79 | \( 1 + 0.285T + 79T^{2} \) |
| 83 | \( 1 - 11.9T + 83T^{2} \) |
| 89 | \( 1 + 2.01T + 89T^{2} \) |
| 97 | \( 1 - 6.55T + 97T^{2} \) |
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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
−9.701374038919174270442661031834, −8.646444386129235290258477914682, −8.041241316193251307296836734201, −7.69114709713630367966700949511, −6.77113263069635594267433331457, −5.14906760713632270232202531891, −4.04794207511389660312637262566, −2.95675840162826184742609675172, −2.18654100231582988810685617707, 0,
2.18654100231582988810685617707, 2.95675840162826184742609675172, 4.04794207511389660312637262566, 5.14906760713632270232202531891, 6.77113263069635594267433331457, 7.69114709713630367966700949511, 8.041241316193251307296836734201, 8.646444386129235290258477914682, 9.701374038919174270442661031834