| L(s) = 1 | − 2.15·2-s − 0.758·3-s + 2.62·4-s − 3.61·5-s + 1.63·6-s − 3.46·7-s − 1.34·8-s − 2.42·9-s + 7.77·10-s − 2.36·11-s − 1.98·12-s − 5.83·13-s + 7.45·14-s + 2.74·15-s − 2.36·16-s − 1.13·17-s + 5.21·18-s − 19-s − 9.48·20-s + 2.62·21-s + 5.08·22-s − 8.30·23-s + 1.01·24-s + 8.06·25-s + 12.5·26-s + 4.11·27-s − 9.09·28-s + ⋯ |
| L(s) = 1 | − 1.52·2-s − 0.437·3-s + 1.31·4-s − 1.61·5-s + 0.665·6-s − 1.30·7-s − 0.474·8-s − 0.808·9-s + 2.45·10-s − 0.712·11-s − 0.574·12-s − 1.61·13-s + 1.99·14-s + 0.707·15-s − 0.590·16-s − 0.276·17-s + 1.22·18-s − 0.229·19-s − 2.12·20-s + 0.573·21-s + 1.08·22-s − 1.73·23-s + 0.207·24-s + 1.61·25-s + 2.46·26-s + 0.791·27-s − 1.71·28-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 | \( 1 + 2.15T + 2T^{2} \) |
| 3 | \( 1 + 0.758T + 3T^{2} \) |
| 5 | \( 1 + 3.61T + 5T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 11 | \( 1 + 2.36T + 11T^{2} \) |
| 13 | \( 1 + 5.83T + 13T^{2} \) |
| 17 | \( 1 + 1.13T + 17T^{2} \) |
| 23 | \( 1 + 8.30T + 23T^{2} \) |
| 29 | \( 1 + 0.468T + 29T^{2} \) |
| 31 | \( 1 - 5.31T + 31T^{2} \) |
| 37 | \( 1 + 8.64T + 37T^{2} \) |
| 41 | \( 1 - 0.0804T + 41T^{2} \) |
| 43 | \( 1 - 3.27T + 43T^{2} \) |
| 47 | \( 1 - 6.11T + 47T^{2} \) |
| 53 | \( 1 - 5.60T + 53T^{2} \) |
| 59 | \( 1 + 2.48T + 59T^{2} \) |
| 61 | \( 1 - 3.79T + 61T^{2} \) |
| 67 | \( 1 + 7.78T + 67T^{2} \) |
| 71 | \( 1 - 8.94T + 71T^{2} \) |
| 73 | \( 1 + 3.10T + 73T^{2} \) |
| 79 | \( 1 + 8.07T + 79T^{2} \) |
| 83 | \( 1 - 4.70T + 83T^{2} \) |
| 89 | \( 1 + 8.02T + 89T^{2} \) |
| 97 | \( 1 - 4.78T + 97T^{2} \) |
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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
−7.83533134716034870540870368733, −7.23231078193148737307613370030, −6.69750505939230794282290157202, −5.79965844989337401087388934686, −4.79336301465508807000927474240, −3.98058932309883272239574660501, −2.96364179445709847470952389404, −2.30329907540560235577595510707, −0.47262923260859783796219146982, 0,
0.47262923260859783796219146982, 2.30329907540560235577595510707, 2.96364179445709847470952389404, 3.98058932309883272239574660501, 4.79336301465508807000927474240, 5.79965844989337401087388934686, 6.69750505939230794282290157202, 7.23231078193148737307613370030, 7.83533134716034870540870368733
