| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s − 8.00·5-s + 6·6-s + 33.5·7-s − 8·8-s + 9·9-s + 16.0·10-s − 58.4·11-s − 12·12-s + 11.5·13-s − 67.0·14-s + 24.0·15-s + 16·16-s − 0.620·17-s − 18·18-s + 62.1·19-s − 32.0·20-s − 100.·21-s + 116.·22-s − 67.7·23-s + 24·24-s − 60.9·25-s − 23.0·26-s − 27·27-s + 134.·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.716·5-s + 0.408·6-s + 1.81·7-s − 0.353·8-s + 0.333·9-s + 0.506·10-s − 1.60·11-s − 0.288·12-s + 0.246·13-s − 1.28·14-s + 0.413·15-s + 0.250·16-s − 0.00885·17-s − 0.235·18-s + 0.749·19-s − 0.358·20-s − 1.04·21-s + 1.13·22-s − 0.614·23-s + 0.204·24-s − 0.487·25-s − 0.174·26-s − 0.192·27-s + 0.905·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 786 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 786 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 3T \) |
| 131 | \( 1 - 131T \) |
| good | 5 | \( 1 + 8.00T + 125T^{2} \) |
| 7 | \( 1 - 33.5T + 343T^{2} \) |
| 11 | \( 1 + 58.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 11.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 0.620T + 4.91e3T^{2} \) |
| 19 | \( 1 - 62.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 67.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 48.0T + 2.43e4T^{2} \) |
| 31 | \( 1 + 177.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 258.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 120.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 439.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 493.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 154.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 660.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 296.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 281.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.15e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.13e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.36e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.28e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 290.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.78e3T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.530998220325183111057711051313, −8.261217646402344567784363512915, −7.88401207365385357843715716936, −7.29517560628967331886065656318, −5.79465321400163757484811657969, −5.08775948932592876388393655535, −4.09377679188118446059486530872, −2.49074239361508863580163376416, −1.28471700998143382109865750817, 0,
1.28471700998143382109865750817, 2.49074239361508863580163376416, 4.09377679188118446059486530872, 5.08775948932592876388393655535, 5.79465321400163757484811657969, 7.29517560628967331886065656318, 7.88401207365385357843715716936, 8.261217646402344567784363512915, 9.530998220325183111057711051313
