L(s) = 1 | + 2-s − 1.30·3-s + 4-s − 5-s − 1.30·6-s + 4.54·7-s + 8-s − 1.28·9-s − 10-s + 1.13·11-s − 1.30·12-s + 1.30·13-s + 4.54·14-s + 1.30·15-s + 16-s − 5.70·17-s − 1.28·18-s − 0.0584·19-s − 20-s − 5.94·21-s + 1.13·22-s − 3.59·23-s − 1.30·24-s + 25-s + 1.30·26-s + 5.60·27-s + 4.54·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.755·3-s + 0.5·4-s − 0.447·5-s − 0.533·6-s + 1.71·7-s + 0.353·8-s − 0.429·9-s − 0.316·10-s + 0.342·11-s − 0.377·12-s + 0.361·13-s + 1.21·14-s + 0.337·15-s + 0.250·16-s − 1.38·17-s − 0.304·18-s − 0.0134·19-s − 0.223·20-s − 1.29·21-s + 0.241·22-s − 0.749·23-s − 0.266·24-s + 0.200·25-s + 0.255·26-s + 1.07·27-s + 0.859·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 601 | \( 1 - T \) |
good | 3 | \( 1 + 1.30T + 3T^{2} \) |
| 7 | \( 1 - 4.54T + 7T^{2} \) |
| 11 | \( 1 - 1.13T + 11T^{2} \) |
| 13 | \( 1 - 1.30T + 13T^{2} \) |
| 17 | \( 1 + 5.70T + 17T^{2} \) |
| 19 | \( 1 + 0.0584T + 19T^{2} \) |
| 23 | \( 1 + 3.59T + 23T^{2} \) |
| 29 | \( 1 + 8.50T + 29T^{2} \) |
| 31 | \( 1 + 0.0788T + 31T^{2} \) |
| 37 | \( 1 + 6.71T + 37T^{2} \) |
| 41 | \( 1 + 6.64T + 41T^{2} \) |
| 43 | \( 1 + 9.54T + 43T^{2} \) |
| 47 | \( 1 + 9.37T + 47T^{2} \) |
| 53 | \( 1 - 5.17T + 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 - 15.5T + 61T^{2} \) |
| 67 | \( 1 - 6.23T + 67T^{2} \) |
| 71 | \( 1 + 2.26T + 71T^{2} \) |
| 73 | \( 1 - 7.12T + 73T^{2} \) |
| 79 | \( 1 + 6.46T + 79T^{2} \) |
| 83 | \( 1 + 5.70T + 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 + 6.04T + 97T^{2} \) |
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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
−7.65646517411972231042683938147, −6.83776526975295743174211281081, −6.25607357043010245300481223715, −5.26270132564272615620638684119, −5.03764791416549581154197703915, −4.18208204092625794433766723619, −3.50790345855463315533246248046, −2.19020223206050195227249382421, −1.51592712745743956736224636748, 0,
1.51592712745743956736224636748, 2.19020223206050195227249382421, 3.50790345855463315533246248046, 4.18208204092625794433766723619, 5.03764791416549581154197703915, 5.26270132564272615620638684119, 6.25607357043010245300481223715, 6.83776526975295743174211281081, 7.65646517411972231042683938147