| L(s) = 1 | − 8.31·2-s + 15.2·3-s + 37.0·4-s + 25·5-s − 126.·6-s + 144.·7-s − 42.3·8-s − 9.68·9-s − 207.·10-s + 121·11-s + 566.·12-s + 149.·13-s − 1.20e3·14-s + 381.·15-s − 834.·16-s − 452.·17-s + 80.5·18-s + 361·19-s + 927.·20-s + 2.20e3·21-s − 1.00e3·22-s − 1.57e3·23-s − 647.·24-s + 625·25-s − 1.24e3·26-s − 3.85e3·27-s + 5.36e3·28-s + ⋯ |
| L(s) = 1 | − 1.46·2-s + 0.979·3-s + 1.15·4-s + 0.447·5-s − 1.43·6-s + 1.11·7-s − 0.234·8-s − 0.0398·9-s − 0.657·10-s + 0.301·11-s + 1.13·12-s + 0.245·13-s − 1.63·14-s + 0.438·15-s − 0.815·16-s − 0.379·17-s + 0.0585·18-s + 0.229·19-s + 0.518·20-s + 1.09·21-s − 0.443·22-s − 0.621·23-s − 0.229·24-s + 0.200·25-s − 0.360·26-s − 1.01·27-s + 1.29·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.876009533\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.876009533\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 - 361T \) |
| good | 2 | \( 1 + 8.31T + 32T^{2} \) |
| 3 | \( 1 - 15.2T + 243T^{2} \) |
| 7 | \( 1 - 144.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 149.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 452.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 1.57e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.58e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.52e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.39e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 4.22e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.58e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.21e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.44e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.97e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.77e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.24e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.41e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.46e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 9.63e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.64e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.05e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 2.55e4T + 8.58e9T^{2} \) |
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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
−9.080779187203271188878279256727, −8.438438900544740038654593074738, −7.923134922421339507358138233549, −7.15345684869585515721249639217, −6.02004582245004503188550851241, −4.84038502493803003057817095929, −3.67931472655228187221136752970, −2.30162136337090220682563590610, −1.80244507703818963572309383154, −0.71527450103181445538246229906,
0.71527450103181445538246229906, 1.80244507703818963572309383154, 2.30162136337090220682563590610, 3.67931472655228187221136752970, 4.84038502493803003057817095929, 6.02004582245004503188550851241, 7.15345684869585515721249639217, 7.923134922421339507358138233549, 8.438438900544740038654593074738, 9.080779187203271188878279256727
