| L(s) = 1 | − 1.82·2-s − 4.65·4-s − 5·5-s − 7·7-s + 23.1·8-s + 9.14·10-s + 64.5·11-s − 32.3·13-s + 12.7·14-s − 5.05·16-s + 56.3·17-s − 2.74·19-s + 23.2·20-s − 118.·22-s − 88.1·23-s + 25·25-s + 59.1·26-s + 32.5·28-s − 246.·29-s − 110.·31-s − 175.·32-s − 103.·34-s + 35·35-s + 120.·37-s + 5.01·38-s − 115.·40-s + 176.·41-s + ⋯ |
| L(s) = 1 | − 0.646·2-s − 0.582·4-s − 0.447·5-s − 0.377·7-s + 1.02·8-s + 0.289·10-s + 1.76·11-s − 0.690·13-s + 0.244·14-s − 0.0790·16-s + 0.803·17-s − 0.0331·19-s + 0.260·20-s − 1.14·22-s − 0.799·23-s + 0.200·25-s + 0.446·26-s + 0.220·28-s − 1.57·29-s − 0.642·31-s − 0.971·32-s − 0.519·34-s + 0.169·35-s + 0.536·37-s + 0.0214·38-s − 0.457·40-s + 0.671·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 + 1.82T + 8T^{2} \) |
| 11 | \( 1 - 64.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 32.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 56.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 2.74T + 6.85e3T^{2} \) |
| 23 | \( 1 + 88.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 246.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 110.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 120.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 176.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 443.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 345.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 260.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 628.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 115.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 951.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 356.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 656.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 440.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 54.4T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.01e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 724.T + 9.12e5T^{2} \) |
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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
−10.58184402270117112180228495246, −9.513269766894992568374061571147, −9.147187989608741493795246390396, −7.940027469793387468935475092495, −7.14606179275406043849722986945, −5.85706181459430471799933668652, −4.43570930068180126537677482430, −3.55980383521730591711847740597, −1.48522751218313573985147771566, 0,
1.48522751218313573985147771566, 3.55980383521730591711847740597, 4.43570930068180126537677482430, 5.85706181459430471799933668652, 7.14606179275406043849722986945, 7.940027469793387468935475092495, 9.147187989608741493795246390396, 9.513269766894992568374061571147, 10.58184402270117112180228495246
