| L(s) = 1 | + (−0.540 − 0.841i)2-s + (−1.59 − 0.229i)3-s + (−0.415 + 0.909i)4-s + (0.883 − 2.05i)5-s + (0.671 + 1.46i)6-s + (−1.35 + 1.17i)7-s + (0.989 − 0.142i)8-s + (−0.374 − 0.109i)9-s + (−2.20 + 0.366i)10-s + (−4.81 − 3.09i)11-s + (0.873 − 1.35i)12-s + (1.68 + 1.45i)13-s + (1.71 + 0.504i)14-s + (−1.88 + 3.08i)15-s + (−0.654 − 0.755i)16-s + (−3.28 + 1.50i)17-s + ⋯ |
| L(s) = 1 | + (−0.382 − 0.594i)2-s + (−0.923 − 0.132i)3-s + (−0.207 + 0.454i)4-s + (0.395 − 0.918i)5-s + (0.273 + 0.599i)6-s + (−0.511 + 0.442i)7-s + (0.349 − 0.0503i)8-s + (−0.124 − 0.0366i)9-s + (−0.697 + 0.116i)10-s + (−1.45 − 0.932i)11-s + (0.252 − 0.392i)12-s + (0.466 + 0.404i)13-s + (0.458 + 0.134i)14-s + (−0.486 + 0.795i)15-s + (−0.163 − 0.188i)16-s + (−0.797 + 0.364i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.924 - 0.380i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.924 - 0.380i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0422642 + 0.213572i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0422642 + 0.213572i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.540 + 0.841i)T \) |
| 5 | \( 1 + (-0.883 + 2.05i)T \) |
| 23 | \( 1 + (4.17 + 2.36i)T \) |
| good | 3 | \( 1 + (1.59 + 0.229i)T + (2.87 + 0.845i)T^{2} \) |
| 7 | \( 1 + (1.35 - 1.17i)T + (0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (4.81 + 3.09i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-1.68 - 1.45i)T + (1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (3.28 - 1.50i)T + (11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (3.15 - 6.90i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (0.955 + 2.09i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (1.29 + 9.02i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (-1.17 + 3.99i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-5.64 + 1.65i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-2.25 - 0.324i)T + (41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + 3.81iT - 47T^{2} \) |
| 53 | \( 1 + (-2.96 + 2.57i)T + (7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (-6.34 + 7.32i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (0.721 + 5.01i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (-2.00 - 3.12i)T + (-27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (0.655 - 0.421i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (13.8 + 6.32i)T + (47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (6.94 - 8.01i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (4.32 - 14.7i)T + (-69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (-0.0331 + 0.230i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (3.29 + 11.2i)T + (-81.6 + 52.4i)T^{2} \) |
| 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
−11.59925524399849097453293722749, −10.79762719696573622953544712365, −9.883365124222143405694680469744, −8.727534621987154407287496051228, −8.060973692052219718171505366012, −6.08386026784156753400500731800, −5.68038506459254831193704325834, −4.10345109503634920722987529238, −2.26094217927654981566899443690, −0.20899647138042545927270518474,
2.67613378200948172776763938066, 4.70805385210310539975909073814, 5.74885498648332823881927829953, 6.72175848567906903073332288907, 7.45111277481664735723005105647, 8.870224338853607418614726407367, 10.20563642455313545789275521271, 10.55860011851821647959713808143, 11.44051971406131332530230809714, 12.94314352343747230877465315043
