| L(s) = 1 | − 8.31·2-s + 11.6·3-s − 58.8·4-s + 148.·5-s − 97.2·6-s + 122.·7-s + 1.55e3·8-s − 2.05e3·9-s − 1.23e3·10-s + 5.63e3·11-s − 688.·12-s − 7.73e3·13-s − 1.01e3·14-s + 1.73e3·15-s − 5.38e3·16-s + 1.71e3·17-s + 1.70e4·18-s − 5.11e4·19-s − 8.71e3·20-s + 1.42e3·21-s − 4.68e4·22-s − 7.70e4·23-s + 1.81e4·24-s − 5.61e4·25-s + 6.43e4·26-s − 4.95e4·27-s − 7.19e3·28-s + ⋯ |
| L(s) = 1 | − 0.734·2-s + 0.250·3-s − 0.459·4-s + 0.529·5-s − 0.183·6-s + 0.134·7-s + 1.07·8-s − 0.937·9-s − 0.389·10-s + 1.27·11-s − 0.115·12-s − 0.976·13-s − 0.0989·14-s + 0.132·15-s − 0.328·16-s + 0.0845·17-s + 0.688·18-s − 1.71·19-s − 0.243·20-s + 0.0336·21-s − 0.938·22-s − 1.32·23-s + 0.268·24-s − 0.719·25-s + 0.717·26-s − 0.484·27-s − 0.0619·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 - 7.95e4T \) |
| good | 2 | \( 1 + 8.31T + 128T^{2} \) |
| 3 | \( 1 - 11.6T + 2.18e3T^{2} \) |
| 5 | \( 1 - 148.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 122.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 5.63e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 7.73e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.71e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 5.11e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 7.70e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 4.80e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.92e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.61e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 6.88e5T + 1.94e11T^{2} \) |
| 47 | \( 1 + 6.19e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 5.74e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.01e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 3.23e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.15e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.46e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.02e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.63e5T + 1.92e13T^{2} \) |
| 83 | \( 1 + 5.49e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 8.18e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.20e7T + 8.07e13T^{2} \) |
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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
−14.13033880772185873700829118294, −12.74002605642983596377774362916, −11.24921951609667383299749752590, −9.809959691077780670166448569638, −9.004675748527173833010086597253, −7.83708027940230810939653257226, −6.03238728490822145036137611994, −4.22003505495745856128667345440, −1.96459362656104226334755752141, 0,
1.96459362656104226334755752141, 4.22003505495745856128667345440, 6.03238728490822145036137611994, 7.83708027940230810939653257226, 9.004675748527173833010086597253, 9.809959691077780670166448569638, 11.24921951609667383299749752590, 12.74002605642983596377774362916, 14.13033880772185873700829118294
