| L(s) = 1 | + 7.82·2-s − 16.3·3-s + 29.2·4-s − 25·5-s − 127.·6-s − 125.·7-s − 21.7·8-s + 22.8·9-s − 195.·10-s − 121·11-s − 476.·12-s + 532.·13-s − 981.·14-s + 407.·15-s − 1.10e3·16-s − 1.37e3·17-s + 178.·18-s − 554.·19-s − 730.·20-s + 2.04e3·21-s − 946.·22-s + 4.25e3·23-s + 353.·24-s + 625·25-s + 4.16e3·26-s + 3.58e3·27-s − 3.66e3·28-s + ⋯ |
| L(s) = 1 | + 1.38·2-s − 1.04·3-s + 0.913·4-s − 0.447·5-s − 1.44·6-s − 0.967·7-s − 0.119·8-s + 0.0939·9-s − 0.618·10-s − 0.301·11-s − 0.955·12-s + 0.873·13-s − 1.33·14-s + 0.467·15-s − 1.07·16-s − 1.15·17-s + 0.129·18-s − 0.352·19-s − 0.408·20-s + 1.01·21-s − 0.417·22-s + 1.67·23-s + 0.125·24-s + 0.200·25-s + 1.20·26-s + 0.947·27-s − 0.883·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 + 121T \) |
| good | 2 | \( 1 - 7.82T + 32T^{2} \) |
| 3 | \( 1 + 16.3T + 243T^{2} \) |
| 7 | \( 1 + 125.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 532.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.37e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 554.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.25e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.97e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.13e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.38e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 679.T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.72e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.51e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 9.54e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.75e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.05e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.87e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.25e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.37e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.86e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.22e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.15e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 7.90e4T + 8.58e9T^{2} \) |
| 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
−13.32779602285801649924992840541, −12.83942875742554205581132419048, −11.59239281058810462992628757457, −10.87011372956552610271840328524, −8.970357153050841115564913238364, −6.78443238236780676041348362223, −5.90571861093459695816930268547, −4.63328584143517183668702116175, −3.18377717240491839984000138461, 0,
3.18377717240491839984000138461, 4.63328584143517183668702116175, 5.90571861093459695816930268547, 6.78443238236780676041348362223, 8.970357153050841115564913238364, 10.87011372956552610271840328524, 11.59239281058810462992628757457, 12.83942875742554205581132419048, 13.32779602285801649924992840541
