| L(s) = 1 | + (−1.47 − 1.16i)2-s + (−0.814 − 0.580i)3-s + (0.362 + 1.49i)4-s + (0.431 − 0.0832i)5-s + (0.529 + 1.80i)6-s + (1.82 − 1.91i)7-s + (−0.361 + 0.790i)8-s + (0.327 + 0.945i)9-s + (−0.735 − 0.378i)10-s + (2.95 + 3.76i)11-s + (0.571 − 1.42i)12-s + (−0.0227 − 0.0354i)13-s + (−4.92 + 0.709i)14-s + (−0.399 − 0.182i)15-s + (4.18 − 2.15i)16-s + (4.38 + 4.18i)17-s + ⋯ |
| L(s) = 1 | + (−1.04 − 0.822i)2-s + (−0.470 − 0.334i)3-s + (0.181 + 0.747i)4-s + (0.193 − 0.0372i)5-s + (0.216 + 0.736i)6-s + (0.689 − 0.723i)7-s + (−0.127 + 0.279i)8-s + (0.109 + 0.315i)9-s + (−0.232 − 0.119i)10-s + (0.891 + 1.13i)11-s + (0.164 − 0.412i)12-s + (−0.00632 − 0.00983i)13-s + (−1.31 + 0.189i)14-s + (−0.103 − 0.0471i)15-s + (1.04 − 0.539i)16-s + (1.06 + 1.01i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.377 + 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.377 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.680084 - 0.457019i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.680084 - 0.457019i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.814 + 0.580i)T \) |
| 7 | \( 1 + (-1.82 + 1.91i)T \) |
| 23 | \( 1 + (2.18 - 4.27i)T \) |
| good | 2 | \( 1 + (1.47 + 1.16i)T + (0.471 + 1.94i)T^{2} \) |
| 5 | \( 1 + (-0.431 + 0.0832i)T + (4.64 - 1.85i)T^{2} \) |
| 11 | \( 1 + (-2.95 - 3.76i)T + (-2.59 + 10.6i)T^{2} \) |
| 13 | \( 1 + (0.0227 + 0.0354i)T + (-5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (-4.38 - 4.18i)T + (0.808 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-1.72 + 1.64i)T + (0.904 - 18.9i)T^{2} \) |
| 29 | \( 1 + (-8.33 + 2.44i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (0.0526 + 0.551i)T + (-30.4 + 5.86i)T^{2} \) |
| 37 | \( 1 + (-4.97 + 1.72i)T + (29.0 - 22.8i)T^{2} \) |
| 41 | \( 1 + (-5.73 + 4.96i)T + (5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (-3.61 + 1.65i)T + (28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + (11.2 + 6.50i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (13.9 + 0.662i)T + (52.7 + 5.03i)T^{2} \) |
| 59 | \( 1 + (-1.41 + 2.74i)T + (-34.2 - 48.0i)T^{2} \) |
| 61 | \( 1 + (-4.84 - 6.80i)T + (-19.9 + 57.6i)T^{2} \) |
| 67 | \( 1 + (0.183 + 0.458i)T + (-48.4 + 46.2i)T^{2} \) |
| 71 | \( 1 + (0.420 - 2.92i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-1.96 + 0.475i)T + (64.8 - 33.4i)T^{2} \) |
| 79 | \( 1 + (-4.99 + 0.238i)T + (78.6 - 7.50i)T^{2} \) |
| 83 | \( 1 + (-4.93 + 5.69i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-11.3 - 1.08i)T + (87.3 + 16.8i)T^{2} \) |
| 97 | \( 1 + (-3.83 - 4.43i)T + (-13.8 + 96.0i)T^{2} \) |
| 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
−10.69033646697185579359247159830, −9.981593774348775444363970833579, −9.369310417100884960122829703554, −8.066341632707521583060006186160, −7.53664637348751165212141743063, −6.27585575050472862881224245212, −5.08790005928657637283718671904, −3.81178899879130085999835996634, −1.99068223512266233154196636799, −1.12539899915974017431074636972,
1.03641872821568402176247203701, 3.17462362134081372747311258651, 4.71562714504122973504040072600, 5.95350230981952100895806226637, 6.39707364457336862608607587433, 7.80823531843108810759643862257, 8.358584766669716422561687653386, 9.358319090181788428945754682943, 9.881812681887204850192610092635, 11.06487915619420932438637429704
