L(s) = 1 | − 2-s + 0.562·3-s + 4-s − 3.89·5-s − 0.562·6-s − 0.867·7-s − 8-s − 2.68·9-s + 3.89·10-s + 0.507·11-s + 0.562·12-s − 2.74·13-s + 0.867·14-s − 2.18·15-s + 16-s + 3.74·17-s + 2.68·18-s + 4.55·19-s − 3.89·20-s − 0.488·21-s − 0.507·22-s − 23-s − 0.562·24-s + 10.1·25-s + 2.74·26-s − 3.19·27-s − 0.867·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.325·3-s + 0.5·4-s − 1.73·5-s − 0.229·6-s − 0.327·7-s − 0.353·8-s − 0.894·9-s + 1.23·10-s + 0.152·11-s + 0.162·12-s − 0.760·13-s + 0.231·14-s − 0.565·15-s + 0.250·16-s + 0.908·17-s + 0.632·18-s + 1.04·19-s − 0.869·20-s − 0.106·21-s − 0.108·22-s − 0.208·23-s − 0.114·24-s + 2.02·25-s + 0.537·26-s − 0.615·27-s − 0.163·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 + T \) |
good | 3 | \( 1 - 0.562T + 3T^{2} \) |
| 5 | \( 1 + 3.89T + 5T^{2} \) |
| 7 | \( 1 + 0.867T + 7T^{2} \) |
| 11 | \( 1 - 0.507T + 11T^{2} \) |
| 13 | \( 1 + 2.74T + 13T^{2} \) |
| 17 | \( 1 - 3.74T + 17T^{2} \) |
| 19 | \( 1 - 4.55T + 19T^{2} \) |
| 29 | \( 1 - 5.93T + 29T^{2} \) |
| 31 | \( 1 + 1.85T + 31T^{2} \) |
| 37 | \( 1 + 0.608T + 37T^{2} \) |
| 41 | \( 1 + 9.86T + 41T^{2} \) |
| 43 | \( 1 + 1.26T + 43T^{2} \) |
| 47 | \( 1 - 8.01T + 47T^{2} \) |
| 53 | \( 1 - 4.96T + 53T^{2} \) |
| 59 | \( 1 - 5.28T + 59T^{2} \) |
| 61 | \( 1 - 13.9T + 61T^{2} \) |
| 67 | \( 1 + 3.14T + 67T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 73 | \( 1 + 9.43T + 73T^{2} \) |
| 79 | \( 1 + 9.73T + 79T^{2} \) |
| 83 | \( 1 + 3.20T + 83T^{2} \) |
| 89 | \( 1 - 2.82T + 89T^{2} \) |
| 97 | \( 1 - 5.75T + 97T^{2} \) |
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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
−7.71258493514740288922297882356, −7.34709487106035839550943089299, −6.62186284011011733429152763571, −5.56236334580349600413859451947, −4.82851815115110899773102738580, −3.69627578565013534792409936546, −3.28134668427608410800203861060, −2.46999324053389376668107268851, −0.964329122152570059481869230684, 0,
0.964329122152570059481869230684, 2.46999324053389376668107268851, 3.28134668427608410800203861060, 3.69627578565013534792409936546, 4.82851815115110899773102738580, 5.56236334580349600413859451947, 6.62186284011011733429152763571, 7.34709487106035839550943089299, 7.71258493514740288922297882356