| L(s) = 1 | − 2-s + 4-s + 0.384·5-s + 0.948·7-s − 8-s − 0.384·10-s − 1.61·11-s − 4.33·13-s − 0.948·14-s + 16-s − 1.49·17-s + 6.66·19-s + 0.384·20-s + 1.61·22-s − 6.58·23-s − 4.85·25-s + 4.33·26-s + 0.948·28-s − 10.1·29-s + 4.20·31-s − 32-s + 1.49·34-s + 0.365·35-s − 3.40·37-s − 6.66·38-s − 0.384·40-s + 3.84·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.172·5-s + 0.358·7-s − 0.353·8-s − 0.121·10-s − 0.485·11-s − 1.20·13-s − 0.253·14-s + 0.250·16-s − 0.362·17-s + 1.52·19-s + 0.0860·20-s + 0.343·22-s − 1.37·23-s − 0.970·25-s + 0.850·26-s + 0.179·28-s − 1.87·29-s + 0.755·31-s − 0.176·32-s + 0.256·34-s + 0.0617·35-s − 0.559·37-s − 1.08·38-s − 0.0608·40-s + 0.600·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1458 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1458 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 0.384T + 5T^{2} \) |
| 7 | \( 1 - 0.948T + 7T^{2} \) |
| 11 | \( 1 + 1.61T + 11T^{2} \) |
| 13 | \( 1 + 4.33T + 13T^{2} \) |
| 17 | \( 1 + 1.49T + 17T^{2} \) |
| 19 | \( 1 - 6.66T + 19T^{2} \) |
| 23 | \( 1 + 6.58T + 23T^{2} \) |
| 29 | \( 1 + 10.1T + 29T^{2} \) |
| 31 | \( 1 - 4.20T + 31T^{2} \) |
| 37 | \( 1 + 3.40T + 37T^{2} \) |
| 41 | \( 1 - 3.84T + 41T^{2} \) |
| 43 | \( 1 - 1.82T + 43T^{2} \) |
| 47 | \( 1 + 9.48T + 47T^{2} \) |
| 53 | \( 1 - 2.01T + 53T^{2} \) |
| 59 | \( 1 - 9.00T + 59T^{2} \) |
| 61 | \( 1 - 8.60T + 61T^{2} \) |
| 67 | \( 1 + 11.6T + 67T^{2} \) |
| 71 | \( 1 + 0.440T + 71T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 79 | \( 1 + 13.3T + 79T^{2} \) |
| 83 | \( 1 + 7.98T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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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
−9.360245066970196974720075144280, −8.131316954114803313601622113396, −7.69914269219120191250318047338, −6.92254768220595860648143010548, −5.77655626303438235980467980010, −5.11527214679120998293049656288, −3.88556933956892538638341997233, −2.63552772018596069645321935470, −1.71026421225617121334907129850, 0,
1.71026421225617121334907129850, 2.63552772018596069645321935470, 3.88556933956892538638341997233, 5.11527214679120998293049656288, 5.77655626303438235980467980010, 6.92254768220595860648143010548, 7.69914269219120191250318047338, 8.131316954114803313601622113396, 9.360245066970196974720075144280
