| L(s) = 1 | − 2.74·2-s + 0.350·3-s + 5.55·4-s + 4.31·5-s − 0.964·6-s − 1.95·7-s − 9.77·8-s − 2.87·9-s − 11.8·10-s + 0.406·11-s + 1.94·12-s + 1.40·13-s + 5.37·14-s + 1.51·15-s + 15.7·16-s − 17-s + 7.90·18-s − 1.50·19-s + 23.9·20-s − 0.686·21-s − 1.11·22-s + 5.78·23-s − 3.42·24-s + 13.6·25-s − 3.85·26-s − 2.06·27-s − 10.8·28-s + ⋯ |
| L(s) = 1 | − 1.94·2-s + 0.202·3-s + 2.77·4-s + 1.92·5-s − 0.393·6-s − 0.739·7-s − 3.45·8-s − 0.958·9-s − 3.74·10-s + 0.122·11-s + 0.562·12-s + 0.388·13-s + 1.43·14-s + 0.390·15-s + 3.93·16-s − 0.242·17-s + 1.86·18-s − 0.344·19-s + 5.35·20-s − 0.149·21-s − 0.238·22-s + 1.20·23-s − 0.699·24-s + 2.72·25-s − 0.755·26-s − 0.396·27-s − 2.05·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8880572945\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8880572945\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + T \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 + 2.74T + 2T^{2} \) |
| 3 | \( 1 - 0.350T + 3T^{2} \) |
| 5 | \( 1 - 4.31T + 5T^{2} \) |
| 7 | \( 1 + 1.95T + 7T^{2} \) |
| 11 | \( 1 - 0.406T + 11T^{2} \) |
| 13 | \( 1 - 1.40T + 13T^{2} \) |
| 19 | \( 1 + 1.50T + 19T^{2} \) |
| 23 | \( 1 - 5.78T + 23T^{2} \) |
| 29 | \( 1 - 6.87T + 29T^{2} \) |
| 31 | \( 1 + 1.67T + 31T^{2} \) |
| 37 | \( 1 - 8.97T + 37T^{2} \) |
| 41 | \( 1 - 11.7T + 41T^{2} \) |
| 43 | \( 1 + 6.25T + 43T^{2} \) |
| 53 | \( 1 + 2.13T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 - 1.07T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 - 9.62T + 71T^{2} \) |
| 73 | \( 1 + 2.68T + 73T^{2} \) |
| 79 | \( 1 - 13.5T + 79T^{2} \) |
| 83 | \( 1 + 0.391T + 83T^{2} \) |
| 89 | \( 1 + 2.71T + 89T^{2} \) |
| 97 | \( 1 + 12.3T + 97T^{2} \) |
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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
−9.999139154728567396285744444357, −9.276639195492621222212688040414, −8.970371303622525966330138685670, −8.058304989618351446526213652712, −6.70022280735301124501288466726, −6.34631377304330078978476069647, −5.49746322847542691107629946629, −2.93738898367305916153544216937, −2.35840557698516059626679197886, −1.04537621431768055733388281509,
1.04537621431768055733388281509, 2.35840557698516059626679197886, 2.93738898367305916153544216937, 5.49746322847542691107629946629, 6.34631377304330078978476069647, 6.70022280735301124501288466726, 8.058304989618351446526213652712, 8.970371303622525966330138685670, 9.276639195492621222212688040414, 9.999139154728567396285744444357
