| L(s) = 1 | − 2-s + 3.27·3-s + 4-s − 0.718·5-s − 3.27·6-s − 3.71·7-s − 8-s + 7.71·9-s + 0.718·10-s − 11-s + 3.27·12-s − 1.02·13-s + 3.71·14-s − 2.35·15-s + 16-s − 5.59·17-s − 7.71·18-s − 1.39·19-s − 0.718·20-s − 12.1·21-s + 22-s + 5.66·23-s − 3.27·24-s − 4.48·25-s + 1.02·26-s + 15.4·27-s − 3.71·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.88·3-s + 0.5·4-s − 0.321·5-s − 1.33·6-s − 1.40·7-s − 0.353·8-s + 2.57·9-s + 0.227·10-s − 0.301·11-s + 0.944·12-s − 0.285·13-s + 0.992·14-s − 0.607·15-s + 0.250·16-s − 1.35·17-s − 1.81·18-s − 0.319·19-s − 0.160·20-s − 2.65·21-s + 0.213·22-s + 1.18·23-s − 0.668·24-s − 0.896·25-s + 0.201·26-s + 2.96·27-s − 0.701·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 197 | \( 1 + T \) |
| good | 3 | \( 1 - 3.27T + 3T^{2} \) |
| 5 | \( 1 + 0.718T + 5T^{2} \) |
| 7 | \( 1 + 3.71T + 7T^{2} \) |
| 13 | \( 1 + 1.02T + 13T^{2} \) |
| 17 | \( 1 + 5.59T + 17T^{2} \) |
| 19 | \( 1 + 1.39T + 19T^{2} \) |
| 23 | \( 1 - 5.66T + 23T^{2} \) |
| 29 | \( 1 - 6.63T + 29T^{2} \) |
| 31 | \( 1 + 5.86T + 31T^{2} \) |
| 37 | \( 1 - 2.90T + 37T^{2} \) |
| 41 | \( 1 + 12.5T + 41T^{2} \) |
| 43 | \( 1 - 7.81T + 43T^{2} \) |
| 47 | \( 1 + 4.12T + 47T^{2} \) |
| 53 | \( 1 + 1.96T + 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 + 6.89T + 61T^{2} \) |
| 67 | \( 1 - 1.71T + 67T^{2} \) |
| 71 | \( 1 + 7.49T + 71T^{2} \) |
| 73 | \( 1 + 1.87T + 73T^{2} \) |
| 79 | \( 1 + 8.53T + 79T^{2} \) |
| 83 | \( 1 + 5.15T + 83T^{2} \) |
| 89 | \( 1 + 0.975T + 89T^{2} \) |
| 97 | \( 1 + 16.1T + 97T^{2} \) |
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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
−8.174148029494842957673089518141, −7.40340329521191583908027232334, −6.91349452594658686748732898752, −6.22942437494329191250883130411, −4.73081767353090344695865952782, −3.88662077772430223840593890723, −3.06117061482780745836561904073, −2.63195091984267522462200548131, −1.62715844748839715021289213909, 0,
1.62715844748839715021289213909, 2.63195091984267522462200548131, 3.06117061482780745836561904073, 3.88662077772430223840593890723, 4.73081767353090344695865952782, 6.22942437494329191250883130411, 6.91349452594658686748732898752, 7.40340329521191583908027232334, 8.174148029494842957673089518141
