L(s) = 1 | − 2.07·2-s + 2.32·4-s + 2.51·5-s + 3.31·7-s − 0.668·8-s − 5.22·10-s + 11-s − 4.69·13-s − 6.89·14-s − 3.25·16-s − 2.84·17-s − 7.33·19-s + 5.83·20-s − 2.07·22-s + 2.08·23-s + 1.32·25-s + 9.76·26-s + 7.69·28-s − 0.764·29-s + 2.67·31-s + 8.10·32-s + 5.91·34-s + 8.33·35-s + 2.93·37-s + 15.2·38-s − 1.68·40-s − 3.67·41-s + ⋯ |
L(s) = 1 | − 1.46·2-s + 1.16·4-s + 1.12·5-s + 1.25·7-s − 0.236·8-s − 1.65·10-s + 0.301·11-s − 1.30·13-s − 1.84·14-s − 0.813·16-s − 0.690·17-s − 1.68·19-s + 1.30·20-s − 0.443·22-s + 0.434·23-s + 0.265·25-s + 1.91·26-s + 1.45·28-s − 0.141·29-s + 0.479·31-s + 1.43·32-s + 1.01·34-s + 1.40·35-s + 0.481·37-s + 2.47·38-s − 0.265·40-s − 0.574·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
good | 2 | \( 1 + 2.07T + 2T^{2} \) |
| 5 | \( 1 - 2.51T + 5T^{2} \) |
| 7 | \( 1 - 3.31T + 7T^{2} \) |
| 13 | \( 1 + 4.69T + 13T^{2} \) |
| 17 | \( 1 + 2.84T + 17T^{2} \) |
| 19 | \( 1 + 7.33T + 19T^{2} \) |
| 23 | \( 1 - 2.08T + 23T^{2} \) |
| 29 | \( 1 + 0.764T + 29T^{2} \) |
| 31 | \( 1 - 2.67T + 31T^{2} \) |
| 37 | \( 1 - 2.93T + 37T^{2} \) |
| 41 | \( 1 + 3.67T + 41T^{2} \) |
| 43 | \( 1 + 9.30T + 43T^{2} \) |
| 47 | \( 1 - 0.466T + 47T^{2} \) |
| 53 | \( 1 - 7.82T + 53T^{2} \) |
| 59 | \( 1 - 7.75T + 59T^{2} \) |
| 67 | \( 1 + 4.65T + 67T^{2} \) |
| 71 | \( 1 + 7.30T + 71T^{2} \) |
| 73 | \( 1 + 6.71T + 73T^{2} \) |
| 79 | \( 1 + 6.44T + 79T^{2} \) |
| 83 | \( 1 + 2.42T + 83T^{2} \) |
| 89 | \( 1 - 2.54T + 89T^{2} \) |
| 97 | \( 1 + 11.8T + 97T^{2} \) |
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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
−7.994786935456314010920246779736, −7.07637711248777056400777610132, −6.65477048189212527719275771612, −5.68814127191079765612891572282, −4.81568497749174378106106162347, −4.29968715559961299829668695885, −2.56805743477909560020423430390, −2.02842096142776514906995239552, −1.38474870133931801224469168057, 0,
1.38474870133931801224469168057, 2.02842096142776514906995239552, 2.56805743477909560020423430390, 4.29968715559961299829668695885, 4.81568497749174378106106162347, 5.68814127191079765612891572282, 6.65477048189212527719275771612, 7.07637711248777056400777610132, 7.994786935456314010920246779736