L(s) = 1 | − 6.95·2-s − 9·3-s + 16.3·4-s − 101.·5-s + 62.5·6-s − 73.4·7-s + 109.·8-s + 81·9-s + 706.·10-s + 103.·11-s − 146.·12-s − 956.·13-s + 510.·14-s + 915.·15-s − 1.27e3·16-s − 1.80e3·17-s − 563.·18-s − 790.·19-s − 1.65e3·20-s + 660.·21-s − 718.·22-s − 1.40e3·23-s − 981.·24-s + 7.21e3·25-s + 6.64e3·26-s − 729·27-s − 1.19e3·28-s + ⋯ |
L(s) = 1 | − 1.22·2-s − 0.577·3-s + 0.509·4-s − 1.81·5-s + 0.709·6-s − 0.566·7-s + 0.602·8-s + 0.333·9-s + 2.23·10-s + 0.257·11-s − 0.294·12-s − 1.56·13-s + 0.695·14-s + 1.05·15-s − 1.24·16-s − 1.51·17-s − 0.409·18-s − 0.502·19-s − 0.927·20-s + 0.326·21-s − 0.316·22-s − 0.552·23-s − 0.347·24-s + 2.30·25-s + 1.92·26-s − 0.192·27-s − 0.288·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + 9T \) |
| 157 | \( 1 + 2.46e4T \) |
good | 2 | \( 1 + 6.95T + 32T^{2} \) |
| 5 | \( 1 + 101.T + 3.12e3T^{2} \) |
| 7 | \( 1 + 73.4T + 1.68e4T^{2} \) |
| 11 | \( 1 - 103.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 956.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.80e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 790.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.40e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.76e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.08e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.81e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 622.T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.25e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.29e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 6.24e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.84e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.64e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.63e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.30e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.58e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.47e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.64e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.33e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.42e3T + 8.58e9T^{2} \) |
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
−9.691695611830972300279618047849, −8.881570136187168970220796292299, −7.927773355326384200420527702006, −7.29274273166333012308681475777, −6.50548131860520924243054955516, −4.65793768830025694715295459994, −4.15466732765991322548166469087, −2.48051494822233696840959205330, −0.66242012421065867357577933022, 0,
0.66242012421065867357577933022, 2.48051494822233696840959205330, 4.15466732765991322548166469087, 4.65793768830025694715295459994, 6.50548131860520924243054955516, 7.29274273166333012308681475777, 7.927773355326384200420527702006, 8.881570136187168970220796292299, 9.691695611830972300279618047849