L(s) = 1 | − 10.5·2-s + 79.7·4-s + 45.7·5-s − 1.90·7-s − 504.·8-s − 483.·10-s − 342.·11-s + 540.·13-s + 20.1·14-s + 2.78e3·16-s − 551.·17-s + 1.65e3·19-s + 3.64e3·20-s + 3.62e3·22-s + 1.80e3·23-s − 1.03e3·25-s − 5.71e3·26-s − 152.·28-s − 2.64e3·29-s − 2.43e3·31-s − 1.32e4·32-s + 5.82e3·34-s − 87.2·35-s − 1.16e4·37-s − 1.75e4·38-s − 2.30e4·40-s + 1.36e4·41-s + ⋯ |
L(s) = 1 | − 1.86·2-s + 2.49·4-s + 0.818·5-s − 0.0147·7-s − 2.78·8-s − 1.52·10-s − 0.853·11-s + 0.886·13-s + 0.0274·14-s + 2.71·16-s − 0.462·17-s + 1.05·19-s + 2.04·20-s + 1.59·22-s + 0.711·23-s − 0.329·25-s − 1.65·26-s − 0.0366·28-s − 0.584·29-s − 0.454·31-s − 2.29·32-s + 0.864·34-s − 0.0120·35-s − 1.39·37-s − 1.96·38-s − 2.28·40-s + 1.27·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 59 | \( 1 - 3.48e3T \) |
good | 2 | \( 1 + 10.5T + 32T^{2} \) |
| 5 | \( 1 - 45.7T + 3.12e3T^{2} \) |
| 7 | \( 1 + 1.90T + 1.68e4T^{2} \) |
| 11 | \( 1 + 342.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 540.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 551.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.65e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.80e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.64e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.43e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.16e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.36e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.36e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 5.33e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 9.60e3T + 4.18e8T^{2} \) |
| 61 | \( 1 - 4.25e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.24e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.73e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.18e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.71e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.05e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.03e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.52e5T + 8.58e9T^{2} \) |
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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
−9.502252679044450596180499195560, −8.904852905041425799415649031402, −7.991506439336502894732625834718, −7.18366776267789854280637950452, −6.23725815278697153034749481929, −5.32659459932849825863344972391, −3.25684217591944264282616930817, −2.13046101459553535397582961656, −1.24133575359309023054489946162, 0,
1.24133575359309023054489946162, 2.13046101459553535397582961656, 3.25684217591944264282616930817, 5.32659459932849825863344972391, 6.23725815278697153034749481929, 7.18366776267789854280637950452, 7.991506439336502894732625834718, 8.904852905041425799415649031402, 9.502252679044450596180499195560