| L(s) = 1 | − 2.30·2-s + 2.60·3-s + 3.31·4-s − 2.40·5-s − 6.01·6-s + 3.07·7-s − 3.03·8-s + 3.80·9-s + 5.53·10-s + 0.844·11-s + 8.65·12-s − 3.14·13-s − 7.10·14-s − 6.26·15-s + 0.360·16-s + 6.14·17-s − 8.77·18-s − 19-s − 7.96·20-s + 8.03·21-s − 1.94·22-s − 4.06·23-s − 7.91·24-s + 0.768·25-s + 7.24·26-s + 2.10·27-s + 10.2·28-s + ⋯ |
| L(s) = 1 | − 1.63·2-s + 1.50·3-s + 1.65·4-s − 1.07·5-s − 2.45·6-s + 1.16·7-s − 1.07·8-s + 1.26·9-s + 1.75·10-s + 0.254·11-s + 2.49·12-s − 0.871·13-s − 1.89·14-s − 1.61·15-s + 0.0902·16-s + 1.49·17-s − 2.06·18-s − 0.229·19-s − 1.78·20-s + 1.75·21-s − 0.415·22-s − 0.847·23-s − 1.61·24-s + 0.153·25-s + 1.42·26-s + 0.405·27-s + 1.92·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 19 | \( 1 + T \) |
| 317 | \( 1 + T \) |
| good | 2 | \( 1 + 2.30T + 2T^{2} \) |
| 3 | \( 1 - 2.60T + 3T^{2} \) |
| 5 | \( 1 + 2.40T + 5T^{2} \) |
| 7 | \( 1 - 3.07T + 7T^{2} \) |
| 11 | \( 1 - 0.844T + 11T^{2} \) |
| 13 | \( 1 + 3.14T + 13T^{2} \) |
| 17 | \( 1 - 6.14T + 17T^{2} \) |
| 23 | \( 1 + 4.06T + 23T^{2} \) |
| 29 | \( 1 + 2.51T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 - 10.1T + 37T^{2} \) |
| 41 | \( 1 + 5.50T + 41T^{2} \) |
| 43 | \( 1 + 12.0T + 43T^{2} \) |
| 47 | \( 1 - 2.56T + 47T^{2} \) |
| 53 | \( 1 - 2.70T + 53T^{2} \) |
| 59 | \( 1 + 2.93T + 59T^{2} \) |
| 61 | \( 1 - 0.124T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 - 5.48T + 71T^{2} \) |
| 73 | \( 1 + 7.86T + 73T^{2} \) |
| 79 | \( 1 + 10.0T + 79T^{2} \) |
| 83 | \( 1 - 0.962T + 83T^{2} \) |
| 89 | \( 1 + 8.65T + 89T^{2} \) |
| 97 | \( 1 + 0.130T + 97T^{2} \) |
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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
−7.967477830760117408125869267874, −7.51115466132966885345737974387, −7.05951233757664876726149123669, −5.67639861211420717318170204511, −4.56828558195710077216118965869, −3.81631887693598167429077075152, −2.99663397716267567201742140091, −2.00415194006211309860380463369, −1.43603084764923702161777463095, 0,
1.43603084764923702161777463095, 2.00415194006211309860380463369, 2.99663397716267567201742140091, 3.81631887693598167429077075152, 4.56828558195710077216118965869, 5.67639861211420717318170204511, 7.05951233757664876726149123669, 7.51115466132966885345737974387, 7.967477830760117408125869267874
