| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s + 8.12·5-s − 6·6-s − 7·7-s − 8·8-s + 9·9-s − 16.2·10-s − 41.2·11-s + 12·12-s + 68.4·13-s + 14·14-s + 24.3·15-s + 16·16-s − 121.·17-s − 18·18-s − 19·19-s + 32.5·20-s − 21·21-s + 82.4·22-s − 42.4·23-s − 24·24-s − 58.9·25-s − 136.·26-s + 27·27-s − 28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.726·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.514·10-s − 1.12·11-s + 0.288·12-s + 1.46·13-s + 0.267·14-s + 0.419·15-s + 0.250·16-s − 1.73·17-s − 0.235·18-s − 0.229·19-s + 0.363·20-s − 0.218·21-s + 0.798·22-s − 0.384·23-s − 0.204·24-s − 0.471·25-s − 1.03·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 + 7T \) |
| 19 | \( 1 + 19T \) |
| good | 5 | \( 1 - 8.12T + 125T^{2} \) |
| 11 | \( 1 + 41.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 68.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 121.T + 4.91e3T^{2} \) |
| 23 | \( 1 + 42.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 220.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 47.1T + 2.97e4T^{2} \) |
| 37 | \( 1 + 306.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 282.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 0.956T + 7.95e4T^{2} \) |
| 47 | \( 1 + 474.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 530.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 844.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 139.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 62.3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 205.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 572.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 410.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 748.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 408.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.11e3T + 9.12e5T^{2} \) |
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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
−9.376289139987429385540452383835, −8.581238692964442293746202457306, −8.099237003878115680802638042357, −6.77714207294848814483378896295, −6.25243607065310862990836302739, −5.03315476336828745580914753483, −3.65451738836983510424920867360, −2.53459512640981087937484703585, −1.64308589239130577643405983716, 0,
1.64308589239130577643405983716, 2.53459512640981087937484703585, 3.65451738836983510424920867360, 5.03315476336828745580914753483, 6.25243607065310862990836302739, 6.77714207294848814483378896295, 8.099237003878115680802638042357, 8.581238692964442293746202457306, 9.376289139987429385540452383835
