| L(s) = 1 | − 2.81·3-s − 0.243·5-s + 0.929·7-s + 4.93·9-s + 4.38·11-s − 1.17·13-s + 0.685·15-s + 6.41·17-s + 19-s − 2.61·21-s − 6.17·23-s − 4.94·25-s − 5.46·27-s − 5.16·29-s − 8.29·31-s − 12.3·33-s − 0.226·35-s − 11.1·37-s + 3.29·39-s + 3.11·41-s + 5.53·43-s − 1.20·45-s + 8.81·47-s − 6.13·49-s − 18.0·51-s + 10.2·53-s − 1.06·55-s + ⋯ |
| L(s) = 1 | − 1.62·3-s − 0.108·5-s + 0.351·7-s + 1.64·9-s + 1.32·11-s − 0.324·13-s + 0.177·15-s + 1.55·17-s + 0.229·19-s − 0.571·21-s − 1.28·23-s − 0.988·25-s − 1.05·27-s − 0.959·29-s − 1.49·31-s − 2.15·33-s − 0.0382·35-s − 1.83·37-s + 0.527·39-s + 0.485·41-s + 0.844·43-s − 0.179·45-s + 1.28·47-s − 0.876·49-s − 2.53·51-s + 1.41·53-s − 0.144·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| good | 3 | \( 1 + 2.81T + 3T^{2} \) |
| 5 | \( 1 + 0.243T + 5T^{2} \) |
| 7 | \( 1 - 0.929T + 7T^{2} \) |
| 11 | \( 1 - 4.38T + 11T^{2} \) |
| 13 | \( 1 + 1.17T + 13T^{2} \) |
| 17 | \( 1 - 6.41T + 17T^{2} \) |
| 23 | \( 1 + 6.17T + 23T^{2} \) |
| 29 | \( 1 + 5.16T + 29T^{2} \) |
| 31 | \( 1 + 8.29T + 31T^{2} \) |
| 37 | \( 1 + 11.1T + 37T^{2} \) |
| 41 | \( 1 - 3.11T + 41T^{2} \) |
| 43 | \( 1 - 5.53T + 43T^{2} \) |
| 47 | \( 1 - 8.81T + 47T^{2} \) |
| 53 | \( 1 - 10.2T + 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 - 0.358T + 61T^{2} \) |
| 67 | \( 1 - 11.8T + 67T^{2} \) |
| 71 | \( 1 - 7.27T + 71T^{2} \) |
| 73 | \( 1 + 14.0T + 73T^{2} \) |
| 83 | \( 1 - 6.31T + 83T^{2} \) |
| 89 | \( 1 - 5.87T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−7.40753379975398321351853910763, −7.08218425561319810523555752008, −5.98658406271656252853507528014, −5.73919214419486065027945854583, −5.06016527215786288619040639727, −4.03837796606622158170059979457, −3.63563550499614821659707145878, −1.94825563107722273471001142749, −1.17074265416223335376936033655, 0,
1.17074265416223335376936033655, 1.94825563107722273471001142749, 3.63563550499614821659707145878, 4.03837796606622158170059979457, 5.06016527215786288619040639727, 5.73919214419486065027945854583, 5.98658406271656252853507528014, 7.08218425561319810523555752008, 7.40753379975398321351853910763
