| L(s) = 1 | + (−1.23 − 0.927i)2-s + (−1.46 + 0.547i)3-s + (0.110 + 0.377i)4-s + (1.69 − 1.46i)5-s + (2.32 + 0.683i)6-s + (−4.02 − 0.876i)7-s + (−0.868 + 2.32i)8-s + (−0.409 + 0.354i)9-s + (−3.45 + 0.238i)10-s + (−4.51 − 0.649i)11-s + (−0.369 − 0.493i)12-s + (−0.844 − 3.88i)13-s + (4.17 + 4.82i)14-s + (−1.68 + 3.07i)15-s + (3.89 − 2.50i)16-s + (0.141 + 0.258i)17-s + ⋯ |
| L(s) = 1 | + (−0.875 − 0.655i)2-s + (−0.848 + 0.316i)3-s + (0.0553 + 0.188i)4-s + (0.757 − 0.653i)5-s + (0.950 + 0.278i)6-s + (−1.52 − 0.331i)7-s + (−0.307 + 0.823i)8-s + (−0.136 + 0.118i)9-s + (−1.09 + 0.0754i)10-s + (−1.36 − 0.195i)11-s + (−0.106 − 0.142i)12-s + (−0.234 − 1.07i)13-s + (1.11 + 1.28i)14-s + (−0.435 + 0.793i)15-s + (0.974 − 0.626i)16-s + (0.0342 + 0.0627i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0320i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0320i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00352219 + 0.219418i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00352219 + 0.219418i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.69 + 1.46i)T \) |
| 23 | \( 1 + (0.292 - 4.78i)T \) |
| good | 2 | \( 1 + (1.23 + 0.927i)T + (0.563 + 1.91i)T^{2} \) |
| 3 | \( 1 + (1.46 - 0.547i)T + (2.26 - 1.96i)T^{2} \) |
| 7 | \( 1 + (4.02 + 0.876i)T + (6.36 + 2.90i)T^{2} \) |
| 11 | \( 1 + (4.51 + 0.649i)T + (10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (0.844 + 3.88i)T + (-11.8 + 5.40i)T^{2} \) |
| 17 | \( 1 + (-0.141 - 0.258i)T + (-9.19 + 14.3i)T^{2} \) |
| 19 | \( 1 + (-3.49 + 1.02i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (-0.403 + 1.37i)T + (-24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (2.77 + 6.08i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (-0.220 + 3.08i)T + (-36.6 - 5.26i)T^{2} \) |
| 41 | \( 1 + (6.08 - 7.02i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (-0.482 - 1.29i)T + (-32.4 + 28.1i)T^{2} \) |
| 47 | \( 1 + (-9.04 + 9.04i)T - 47iT^{2} \) |
| 53 | \( 1 + (0.184 - 0.847i)T + (-48.2 - 22.0i)T^{2} \) |
| 59 | \( 1 + (2.33 - 3.63i)T + (-24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (3.31 - 1.51i)T + (39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-0.270 + 0.361i)T + (-18.8 - 64.2i)T^{2} \) |
| 71 | \( 1 + (0.537 + 3.73i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (3.83 + 2.09i)T + (39.4 + 61.4i)T^{2} \) |
| 79 | \( 1 + (8.40 + 5.39i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (-0.939 - 0.0672i)T + (82.1 + 11.8i)T^{2} \) |
| 89 | \( 1 + (-1.99 + 4.35i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (7.77 - 0.556i)T + (96.0 - 13.8i)T^{2} \) |
| 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
−12.98739022728605615932502664518, −11.78526021828480528767783606242, −10.50384148311578455168583570828, −10.10859938293795059572800894404, −9.235081685196203040822623058868, −7.83390659027218753300128879357, −5.86612921384523157521295244970, −5.30010180892064442940163565049, −2.80574410052741156463498113576, −0.32863689174953200078330714399,
3.01932520409192900068422646256, 5.60240408996451414205564154346, 6.56974626535451590138818664523, 7.17812479139612480088898793836, 8.888166347715044326940320093396, 9.776512555257760426313406916505, 10.62466343367965654938539891039, 12.20560057379735038487167603726, 12.86326168598345903728913510960, 14.07739990784167071247466837154
