| L(s) = 1 | + 2-s + 3-s + 4-s − 3.31·5-s + 6-s + 3.84·7-s + 8-s + 9-s − 3.31·10-s + 5.49·11-s + 12-s + 13-s + 3.84·14-s − 3.31·15-s + 16-s + 2·17-s + 18-s + 0.938·19-s − 3.31·20-s + 3.84·21-s + 5.49·22-s + 3.93·23-s + 24-s + 5.96·25-s + 26-s + 27-s + 3.84·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.48·5-s + 0.408·6-s + 1.45·7-s + 0.353·8-s + 0.333·9-s − 1.04·10-s + 1.65·11-s + 0.288·12-s + 0.277·13-s + 1.02·14-s − 0.854·15-s + 0.250·16-s + 0.485·17-s + 0.235·18-s + 0.215·19-s − 0.740·20-s + 0.838·21-s + 1.17·22-s + 0.821·23-s + 0.204·24-s + 1.19·25-s + 0.196·26-s + 0.192·27-s + 0.725·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.695236917\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.695236917\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
| good | 5 | \( 1 + 3.31T + 5T^{2} \) |
| 7 | \( 1 - 3.84T + 7T^{2} \) |
| 11 | \( 1 - 5.49T + 11T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 0.938T + 19T^{2} \) |
| 23 | \( 1 - 3.93T + 23T^{2} \) |
| 29 | \( 1 - 2.02T + 29T^{2} \) |
| 31 | \( 1 + 0.938T + 31T^{2} \) |
| 37 | \( 1 + 7.49T + 37T^{2} \) |
| 41 | \( 1 - 11.6T + 41T^{2} \) |
| 43 | \( 1 - 2.50T + 43T^{2} \) |
| 47 | \( 1 + 5.51T + 47T^{2} \) |
| 53 | \( 1 + 9.60T + 53T^{2} \) |
| 59 | \( 1 - 4.34T + 59T^{2} \) |
| 61 | \( 1 + 1.44T + 61T^{2} \) |
| 67 | \( 1 + 4.24T + 67T^{2} \) |
| 71 | \( 1 + 9.79T + 71T^{2} \) |
| 73 | \( 1 - 2.33T + 73T^{2} \) |
| 79 | \( 1 + 15.5T + 79T^{2} \) |
| 83 | \( 1 + 0.689T + 83T^{2} \) |
| 89 | \( 1 + 2.36T + 89T^{2} \) |
| 97 | \( 1 + 19.1T + 97T^{2} \) |
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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
−7.63476343131034783837211246322, −7.35922479053591472888879496900, −6.56941080306664109569833142721, −5.62230134184147739505637436708, −4.63532730923812091041570715798, −4.33681615770150313892221954657, −3.61981604917821591341934800357, −3.00775012750816932149640105608, −1.71592333224264253606442450300, −1.04954133379681511341564652904,
1.04954133379681511341564652904, 1.71592333224264253606442450300, 3.00775012750816932149640105608, 3.61981604917821591341934800357, 4.33681615770150313892221954657, 4.63532730923812091041570715798, 5.62230134184147739505637436708, 6.56941080306664109569833142721, 7.35922479053591472888879496900, 7.63476343131034783837211246322
