| L(s) = 1 | − 2.77·2-s − 2.37·3-s + 5.67·4-s − 5-s + 6.58·6-s − 3.31·7-s − 10.1·8-s + 2.65·9-s + 2.77·10-s + 11-s − 13.4·12-s + 3.17·13-s + 9.18·14-s + 2.37·15-s + 16.8·16-s − 0.799·17-s − 7.35·18-s − 2.55·19-s − 5.67·20-s + 7.88·21-s − 2.77·22-s − 6.05·23-s + 24.2·24-s + 25-s − 8.80·26-s + 0.824·27-s − 18.8·28-s + ⋯ |
| L(s) = 1 | − 1.95·2-s − 1.37·3-s + 2.83·4-s − 0.447·5-s + 2.68·6-s − 1.25·7-s − 3.59·8-s + 0.884·9-s + 0.876·10-s + 0.301·11-s − 3.89·12-s + 0.881·13-s + 2.45·14-s + 0.613·15-s + 4.21·16-s − 0.193·17-s − 1.73·18-s − 0.586·19-s − 1.26·20-s + 1.72·21-s − 0.590·22-s − 1.26·23-s + 4.94·24-s + 0.200·25-s − 1.72·26-s + 0.158·27-s − 3.55·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| good | 2 | \( 1 + 2.77T + 2T^{2} \) |
| 3 | \( 1 + 2.37T + 3T^{2} \) |
| 7 | \( 1 + 3.31T + 7T^{2} \) |
| 13 | \( 1 - 3.17T + 13T^{2} \) |
| 17 | \( 1 + 0.799T + 17T^{2} \) |
| 19 | \( 1 + 2.55T + 19T^{2} \) |
| 23 | \( 1 + 6.05T + 23T^{2} \) |
| 29 | \( 1 + 7.20T + 29T^{2} \) |
| 31 | \( 1 - 2.29T + 31T^{2} \) |
| 37 | \( 1 + 0.722T + 37T^{2} \) |
| 41 | \( 1 - 1.58T + 41T^{2} \) |
| 43 | \( 1 - 3.98T + 43T^{2} \) |
| 47 | \( 1 - 8.26T + 47T^{2} \) |
| 53 | \( 1 + 7.20T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 + 4.81T + 61T^{2} \) |
| 67 | \( 1 - 1.89T + 67T^{2} \) |
| 71 | \( 1 - 2.23T + 71T^{2} \) |
| 79 | \( 1 + 7.31T + 79T^{2} \) |
| 83 | \( 1 + 4.00T + 83T^{2} \) |
| 89 | \( 1 + 13.5T + 89T^{2} \) |
| 97 | \( 1 - 4.41T + 97T^{2} \) |
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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
−8.219297055635738200743407535547, −7.28604031165880071551998479585, −6.76058341183546265570377555213, −6.00705564190748643406196091726, −5.83070877812672089908455660063, −4.11631960879726067478656114144, −3.17979139546636969188170669860, −1.98408833531039293297778582148, −0.78672257005348506462573555354, 0,
0.78672257005348506462573555354, 1.98408833531039293297778582148, 3.17979139546636969188170669860, 4.11631960879726067478656114144, 5.83070877812672089908455660063, 6.00705564190748643406196091726, 6.76058341183546265570377555213, 7.28604031165880071551998479585, 8.219297055635738200743407535547
