L(s) = 1 | − 2-s − 2.56·3-s + 4-s + 3.54·5-s + 2.56·6-s + 2.20·7-s − 8-s + 3.60·9-s − 3.54·10-s + 2.97·11-s − 2.56·12-s − 5.93·13-s − 2.20·14-s − 9.10·15-s + 16-s − 5.64·17-s − 3.60·18-s − 2.23·19-s + 3.54·20-s − 5.67·21-s − 2.97·22-s − 8.47·23-s + 2.56·24-s + 7.55·25-s + 5.93·26-s − 1.55·27-s + 2.20·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.48·3-s + 0.5·4-s + 1.58·5-s + 1.04·6-s + 0.834·7-s − 0.353·8-s + 1.20·9-s − 1.12·10-s + 0.897·11-s − 0.741·12-s − 1.64·13-s − 0.590·14-s − 2.35·15-s + 0.250·16-s − 1.36·17-s − 0.849·18-s − 0.512·19-s + 0.792·20-s − 1.23·21-s − 0.634·22-s − 1.76·23-s + 0.524·24-s + 1.51·25-s + 1.16·26-s − 0.298·27-s + 0.417·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 751 | \( 1 + T \) |
good | 3 | \( 1 + 2.56T + 3T^{2} \) |
| 5 | \( 1 - 3.54T + 5T^{2} \) |
| 7 | \( 1 - 2.20T + 7T^{2} \) |
| 11 | \( 1 - 2.97T + 11T^{2} \) |
| 13 | \( 1 + 5.93T + 13T^{2} \) |
| 17 | \( 1 + 5.64T + 17T^{2} \) |
| 19 | \( 1 + 2.23T + 19T^{2} \) |
| 23 | \( 1 + 8.47T + 23T^{2} \) |
| 29 | \( 1 + 6.29T + 29T^{2} \) |
| 31 | \( 1 - 2.30T + 31T^{2} \) |
| 37 | \( 1 + 11.4T + 37T^{2} \) |
| 41 | \( 1 - 2.85T + 41T^{2} \) |
| 43 | \( 1 + 3.82T + 43T^{2} \) |
| 47 | \( 1 - 3.68T + 47T^{2} \) |
| 53 | \( 1 + 0.971T + 53T^{2} \) |
| 59 | \( 1 - 2.49T + 59T^{2} \) |
| 61 | \( 1 - 4.55T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 + 15.7T + 71T^{2} \) |
| 73 | \( 1 + 9.88T + 73T^{2} \) |
| 79 | \( 1 - 3.58T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 + 7.96T + 89T^{2} \) |
| 97 | \( 1 + 11.3T + 97T^{2} \) |
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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
−9.315173270264486302608559466070, −8.442279954158582480449402442610, −7.18517274801903311923838331875, −6.58485110098823958150293699848, −5.86607367106501905821519358264, −5.17851019130684037495778635021, −4.33543242033871787445018710167, −2.21703251142069649081973264088, −1.67238335714041377558229799328, 0,
1.67238335714041377558229799328, 2.21703251142069649081973264088, 4.33543242033871787445018710167, 5.17851019130684037495778635021, 5.86607367106501905821519358264, 6.58485110098823958150293699848, 7.18517274801903311923838331875, 8.442279954158582480449402442610, 9.315173270264486302608559466070