| L(s) = 1 | + 2-s − 4-s + 2·5-s − 3·8-s + 2·10-s − 2.47·11-s − 2·13-s − 16-s + 4.47·17-s + 2.47·19-s − 2·20-s − 2.47·22-s + 23-s − 25-s − 2·26-s − 29-s − 8·31-s + 5·32-s + 4.47·34-s + 4.47·37-s + 2.47·38-s − 6·40-s − 6.94·41-s − 2.47·43-s + 2.47·44-s + 46-s + 4.94·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.5·4-s + 0.894·5-s − 1.06·8-s + 0.632·10-s − 0.745·11-s − 0.554·13-s − 0.250·16-s + 1.08·17-s + 0.567·19-s − 0.447·20-s − 0.527·22-s + 0.208·23-s − 0.200·25-s − 0.392·26-s − 0.185·29-s − 1.43·31-s + 0.883·32-s + 0.766·34-s + 0.735·37-s + 0.401·38-s − 0.948·40-s − 1.08·41-s − 0.376·43-s + 0.372·44-s + 0.147·46-s + 0.721·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - T + 2T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 2.47T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 - 2.47T + 19T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 4.47T + 37T^{2} \) |
| 41 | \( 1 + 6.94T + 41T^{2} \) |
| 43 | \( 1 + 2.47T + 43T^{2} \) |
| 47 | \( 1 - 4.94T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 3.52T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 6.94T + 73T^{2} \) |
| 79 | \( 1 - 1.52T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 + 9.41T + 97T^{2} \) |
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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
−7.71182651649597788721055743234, −6.94105790223172752510389053568, −5.96250515086604529027903171350, −5.43382488701646099582535643771, −5.08439916310177821279721254877, −4.08713993306287398362505792700, −3.25301868799606766243297640964, −2.55212796972774549328991505156, −1.43987367473434648678314730374, 0,
1.43987367473434648678314730374, 2.55212796972774549328991505156, 3.25301868799606766243297640964, 4.08713993306287398362505792700, 5.08439916310177821279721254877, 5.43382488701646099582535643771, 5.96250515086604529027903171350, 6.94105790223172752510389053568, 7.71182651649597788721055743234
