| L(s) = 1 | + 2-s + 3-s − 4-s + 6-s − 4.47·7-s − 3·8-s + 9-s − 1.23·11-s − 12-s + 5.61·13-s − 4.47·14-s − 16-s + 3.85·17-s + 18-s + 1.23·19-s − 4.47·21-s − 1.23·22-s + 4.47·23-s − 3·24-s + 5.61·26-s + 27-s + 4.47·28-s + 6.61·29-s + 2.76·31-s + 5·32-s − 1.23·33-s + 3.85·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 0.5·4-s + 0.408·6-s − 1.69·7-s − 1.06·8-s + 0.333·9-s − 0.372·11-s − 0.288·12-s + 1.55·13-s − 1.19·14-s − 0.250·16-s + 0.934·17-s + 0.235·18-s + 0.283·19-s − 0.975·21-s − 0.263·22-s + 0.932·23-s − 0.612·24-s + 1.10·26-s + 0.192·27-s + 0.845·28-s + 1.22·29-s + 0.496·31-s + 0.883·32-s − 0.215·33-s + 0.660·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.158643311\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.158643311\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - T + 2T^{2} \) |
| 7 | \( 1 + 4.47T + 7T^{2} \) |
| 11 | \( 1 + 1.23T + 11T^{2} \) |
| 13 | \( 1 - 5.61T + 13T^{2} \) |
| 17 | \( 1 - 3.85T + 17T^{2} \) |
| 19 | \( 1 - 1.23T + 19T^{2} \) |
| 23 | \( 1 - 4.47T + 23T^{2} \) |
| 29 | \( 1 - 6.61T + 29T^{2} \) |
| 31 | \( 1 - 2.76T + 31T^{2} \) |
| 37 | \( 1 + 3.09T + 37T^{2} \) |
| 41 | \( 1 + 3.61T + 41T^{2} \) |
| 43 | \( 1 - 7.70T + 43T^{2} \) |
| 47 | \( 1 + 0.763T + 47T^{2} \) |
| 53 | \( 1 - 3.61T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 1.61T + 61T^{2} \) |
| 67 | \( 1 - 0.763T + 67T^{2} \) |
| 71 | \( 1 + 5.23T + 71T^{2} \) |
| 73 | \( 1 + 8.09T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 12.4T + 83T^{2} \) |
| 89 | \( 1 - 5.38T + 89T^{2} \) |
| 97 | \( 1 - 2.14T + 97T^{2} \) |
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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
−9.058052298791271090132899876655, −8.717917507663199580099875647359, −7.67936052246176006397150435575, −6.58509042088450208041408405114, −6.05761204268223850290119130945, −5.16098011382932892381786557334, −4.03927744598188054867435992876, −3.31996544183404014002537374828, −2.86783374174408485044327812798, −0.886968208092717484353992052130,
0.886968208092717484353992052130, 2.86783374174408485044327812798, 3.31996544183404014002537374828, 4.03927744598188054867435992876, 5.16098011382932892381786557334, 6.05761204268223850290119130945, 6.58509042088450208041408405114, 7.67936052246176006397150435575, 8.717917507663199580099875647359, 9.058052298791271090132899876655
