| L(s) = 1 | − 4·2-s + 6.72·3-s + 16·4-s + 25·5-s − 26.8·6-s − 229.·7-s − 64·8-s − 197.·9-s − 100·10-s − 676.·11-s + 107.·12-s + 521.·13-s + 917.·14-s + 168.·15-s + 256·16-s + 1.80e3·17-s + 791.·18-s − 1.05e3·19-s + 400·20-s − 1.54e3·21-s + 2.70e3·22-s − 1.89e3·23-s − 430.·24-s + 625·25-s − 2.08e3·26-s − 2.96e3·27-s − 3.66e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.431·3-s + 0.5·4-s + 0.447·5-s − 0.304·6-s − 1.76·7-s − 0.353·8-s − 0.814·9-s − 0.316·10-s − 1.68·11-s + 0.215·12-s + 0.856·13-s + 1.25·14-s + 0.192·15-s + 0.250·16-s + 1.51·17-s + 0.575·18-s − 0.667·19-s + 0.223·20-s − 0.762·21-s + 1.19·22-s − 0.746·23-s − 0.152·24-s + 0.200·25-s − 0.605·26-s − 0.782·27-s − 0.884·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 790 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.5178956906\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5178956906\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 5 | \( 1 - 25T \) |
| 79 | \( 1 + 6.24e3T \) |
| good | 3 | \( 1 - 6.72T + 243T^{2} \) |
| 7 | \( 1 + 229.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 676.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 521.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.80e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.05e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.89e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.76e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.06e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.15e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.74e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.08e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.27e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.22e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 551.T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.39e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.12e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.13e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.57e4T + 2.07e9T^{2} \) |
| 83 | \( 1 + 7.82e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.60e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.80e3T + 8.58e9T^{2} \) |
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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
−9.493547404258561745567992011875, −8.777106922815188763722502434461, −7.960757874822755966350226222463, −7.09252761517201987860269252886, −5.90050412912161724070622874875, −5.61265979420643337968648397301, −3.53768035775655108745421201846, −3.00214647309235305194010458046, −1.97369808706811252271650505707, −0.33314199416243991919696589779,
0.33314199416243991919696589779, 1.97369808706811252271650505707, 3.00214647309235305194010458046, 3.53768035775655108745421201846, 5.61265979420643337968648397301, 5.90050412912161724070622874875, 7.09252761517201987860269252886, 7.960757874822755966350226222463, 8.777106922815188763722502434461, 9.493547404258561745567992011875
