| L(s) = 1 | − 4.35·2-s + 5.79·3-s + 10.9·4-s − 1.94·5-s − 25.2·6-s + 27.7·7-s − 13.0·8-s + 6.62·9-s + 8.46·10-s + 63.7·12-s − 37.7·13-s − 120.·14-s − 11.2·15-s − 31.0·16-s − 17·17-s − 28.8·18-s − 21.1·19-s − 21.3·20-s + 160.·21-s + 35.8·23-s − 75.7·24-s − 121.·25-s + 164.·26-s − 118.·27-s + 304.·28-s + 124.·29-s + 49.0·30-s + ⋯ |
| L(s) = 1 | − 1.54·2-s + 1.11·3-s + 1.37·4-s − 0.173·5-s − 1.71·6-s + 1.49·7-s − 0.577·8-s + 0.245·9-s + 0.267·10-s + 1.53·12-s − 0.804·13-s − 2.30·14-s − 0.193·15-s − 0.485·16-s − 0.242·17-s − 0.377·18-s − 0.255·19-s − 0.238·20-s + 1.66·21-s + 0.325·23-s − 0.644·24-s − 0.969·25-s + 1.23·26-s − 0.842·27-s + 2.05·28-s + 0.797·29-s + 0.298·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2057 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2057 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 \) |
| 17 | \( 1 + 17T \) |
| good | 2 | \( 1 + 4.35T + 8T^{2} \) |
| 3 | \( 1 - 5.79T + 27T^{2} \) |
| 5 | \( 1 + 1.94T + 125T^{2} \) |
| 7 | \( 1 - 27.7T + 343T^{2} \) |
| 13 | \( 1 + 37.7T + 2.19e3T^{2} \) |
| 19 | \( 1 + 21.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 35.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 124.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 207.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 131.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 243.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 21.1T + 7.95e4T^{2} \) |
| 47 | \( 1 + 489.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 683.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 195.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 510.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 836.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 607.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 181.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 758.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 922.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 514.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 656.T + 9.12e5T^{2} \) |
| 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
−8.358527502863991042530333960257, −7.922827734270752782666426040466, −7.43308037317099874569898250259, −6.40537321756531395292323110756, −5.01727846835600386005306788131, −4.27286956617968691928121246075, −2.88424955375921414493169557179, −2.09368878109390491504789327175, −1.33555398352624556788328130817, 0,
1.33555398352624556788328130817, 2.09368878109390491504789327175, 2.88424955375921414493169557179, 4.27286956617968691928121246075, 5.01727846835600386005306788131, 6.40537321756531395292323110756, 7.43308037317099874569898250259, 7.922827734270752782666426040466, 8.358527502863991042530333960257
