| L(s) = 1 | − 1.34·2-s − 3-s − 0.183·4-s + 1.68·5-s + 1.34·6-s + 7-s + 2.94·8-s + 9-s − 2.26·10-s − 0.466·11-s + 0.183·12-s + 5.47·13-s − 1.34·14-s − 1.68·15-s − 3.59·16-s − 1.10·17-s − 1.34·18-s + 3.56·19-s − 0.308·20-s − 21-s + 0.628·22-s − 0.444·23-s − 2.94·24-s − 2.17·25-s − 7.38·26-s − 27-s − 0.183·28-s + ⋯ |
| L(s) = 1 | − 0.952·2-s − 0.577·3-s − 0.0918·4-s + 0.752·5-s + 0.550·6-s + 0.377·7-s + 1.04·8-s + 0.333·9-s − 0.716·10-s − 0.140·11-s + 0.0530·12-s + 1.51·13-s − 0.360·14-s − 0.434·15-s − 0.899·16-s − 0.268·17-s − 0.317·18-s + 0.818·19-s − 0.0690·20-s − 0.218·21-s + 0.133·22-s − 0.0927·23-s − 0.600·24-s − 0.434·25-s − 1.44·26-s − 0.192·27-s − 0.0347·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 383 | \( 1 - T \) |
| good | 2 | \( 1 + 1.34T + 2T^{2} \) |
| 5 | \( 1 - 1.68T + 5T^{2} \) |
| 11 | \( 1 + 0.466T + 11T^{2} \) |
| 13 | \( 1 - 5.47T + 13T^{2} \) |
| 17 | \( 1 + 1.10T + 17T^{2} \) |
| 19 | \( 1 - 3.56T + 19T^{2} \) |
| 23 | \( 1 + 0.444T + 23T^{2} \) |
| 29 | \( 1 - 6.10T + 29T^{2} \) |
| 31 | \( 1 - 2.34T + 31T^{2} \) |
| 37 | \( 1 + 11.6T + 37T^{2} \) |
| 41 | \( 1 - 6.69T + 41T^{2} \) |
| 43 | \( 1 + 9.14T + 43T^{2} \) |
| 47 | \( 1 + 6.86T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 + 6.20T + 59T^{2} \) |
| 61 | \( 1 + 5.83T + 61T^{2} \) |
| 67 | \( 1 + 14.4T + 67T^{2} \) |
| 71 | \( 1 + 2.18T + 71T^{2} \) |
| 73 | \( 1 - 0.501T + 73T^{2} \) |
| 79 | \( 1 - 16.4T + 79T^{2} \) |
| 83 | \( 1 + 14.0T + 83T^{2} \) |
| 89 | \( 1 + 3.04T + 89T^{2} \) |
| 97 | \( 1 + 17.6T + 97T^{2} \) |
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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
−7.68801957946851459764963836821, −6.75059779322018605557950607274, −6.22427458279818021836429299432, −5.41180251604434515182178416761, −4.81885112787536118983694227274, −4.00494653964114999785752353165, −3.00452622016488720353114589108, −1.61417908897337392069124553691, −1.33441352762429679563391151771, 0,
1.33441352762429679563391151771, 1.61417908897337392069124553691, 3.00452622016488720353114589108, 4.00494653964114999785752353165, 4.81885112787536118983694227274, 5.41180251604434515182178416761, 6.22427458279818021836429299432, 6.75059779322018605557950607274, 7.68801957946851459764963836821
