| L(s) = 1 | − 4.80e5i·2-s − 3.45e7i·3-s − 9.33e10·4-s − 1.66e13·6-s − 5.42e15i·7-s − 2.11e16i·8-s + 4.49e17·9-s − 1.03e18·11-s + 3.22e18i·12-s + 1.23e19i·13-s − 2.60e21·14-s − 2.30e22·16-s − 5.41e22i·17-s − 2.15e23i·18-s + 3.08e23·19-s + ⋯ |
| L(s) = 1 | − 1.29i·2-s − 0.0515i·3-s − 0.679·4-s − 0.0667·6-s − 1.25i·7-s − 0.415i·8-s + 0.997·9-s − 0.0560·11-s + 0.0350i·12-s + 0.0303i·13-s − 1.63·14-s − 1.21·16-s − 0.934i·17-s − 1.29i·18-s + 0.680·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(38-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+37/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(19)\) |
\(\approx\) |
\(0.7519829852\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7519829852\) |
| \(L(\frac{39}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 + 4.80e5iT - 1.37e11T^{2} \) |
| 3 | \( 1 + 3.45e7iT - 4.50e17T^{2} \) |
| 7 | \( 1 + 5.42e15iT - 1.85e31T^{2} \) |
| 11 | \( 1 + 1.03e18T + 3.40e38T^{2} \) |
| 13 | \( 1 - 1.23e19iT - 1.64e41T^{2} \) |
| 17 | \( 1 + 5.41e22iT - 3.36e45T^{2} \) |
| 19 | \( 1 - 3.08e23T + 2.06e47T^{2} \) |
| 23 | \( 1 - 2.61e25iT - 2.42e50T^{2} \) |
| 29 | \( 1 + 2.49e26T + 1.28e54T^{2} \) |
| 31 | \( 1 + 2.34e27T + 1.51e55T^{2} \) |
| 37 | \( 1 - 6.21e28iT - 1.05e58T^{2} \) |
| 41 | \( 1 + 8.53e29T + 4.70e59T^{2} \) |
| 43 | \( 1 + 3.53e29iT - 2.74e60T^{2} \) |
| 47 | \( 1 - 6.68e29iT - 7.37e61T^{2} \) |
| 53 | \( 1 + 1.24e32iT - 6.28e63T^{2} \) |
| 59 | \( 1 + 6.57e31T + 3.32e65T^{2} \) |
| 61 | \( 1 + 1.06e33T + 1.14e66T^{2} \) |
| 67 | \( 1 + 8.44e33iT - 3.67e67T^{2} \) |
| 71 | \( 1 + 7.04e33T + 3.13e68T^{2} \) |
| 73 | \( 1 + 9.36e33iT - 8.76e68T^{2} \) |
| 79 | \( 1 + 1.51e35T + 1.63e70T^{2} \) |
| 83 | \( 1 - 1.43e35iT - 1.01e71T^{2} \) |
| 89 | \( 1 + 2.28e35T + 1.34e72T^{2} \) |
| 97 | \( 1 - 7.07e36iT - 3.24e73T^{2} \) |
| 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
−10.08825581779478192659264822838, −9.456908280385528574654022236187, −7.55703101563525520558780414205, −6.86576296954759850702204769466, −4.99141963107884952760491815130, −3.89981183664887117745918828995, −3.19628568628580867001106655234, −1.74569867412075831408991212712, −1.12577152703225214822048256987, −0.12843252446085591193563613650,
1.57527619239360043289703522741, 2.65733701182515049661562570090, 4.27480837362406501285910272327, 5.36258687809282993913279347802, 6.20803092466880470575235927502, 7.20449059359379554291790006760, 8.296543899628476613444789253913, 9.152593387680625702396900212783, 10.54687573063967507740772203098, 11.99518109412815864055970036026
