| L(s) = 1 | + (0.173 + 0.984i)2-s + (−0.0375 − 0.0536i)3-s + (−0.939 + 0.342i)4-s + (0.0386 − 2.23i)5-s + (0.0462 − 0.0462i)6-s + (−4.08 + 0.357i)7-s + (−0.5 − 0.866i)8-s + (1.02 − 2.81i)9-s + (2.20 − 0.350i)10-s + (5.46 − 3.15i)11-s + (0.0536 + 0.0375i)12-s + (−2.82 + 1.02i)13-s + (−1.06 − 3.96i)14-s + (−0.121 + 0.0818i)15-s + (0.766 − 0.642i)16-s + (1.25 − 3.44i)17-s + ⋯ |
| L(s) = 1 | + (0.122 + 0.696i)2-s + (−0.0216 − 0.0309i)3-s + (−0.469 + 0.171i)4-s + (0.0172 − 0.999i)5-s + (0.0188 − 0.0188i)6-s + (−1.54 + 0.135i)7-s + (−0.176 − 0.306i)8-s + (0.341 − 0.938i)9-s + (0.698 − 0.110i)10-s + (1.64 − 0.951i)11-s + (0.0154 + 0.0108i)12-s + (−0.783 + 0.285i)13-s + (−0.283 − 1.05i)14-s + (−0.0313 + 0.0211i)15-s + (0.191 − 0.160i)16-s + (0.304 − 0.836i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.639 + 0.768i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.639 + 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.968554 - 0.453860i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.968554 - 0.453860i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.173 - 0.984i)T \) |
| 5 | \( 1 + (-0.0386 + 2.23i)T \) |
| 37 | \( 1 + (3.43 + 5.02i)T \) |
| good | 3 | \( 1 + (0.0375 + 0.0536i)T + (-1.02 + 2.81i)T^{2} \) |
| 7 | \( 1 + (4.08 - 0.357i)T + (6.89 - 1.21i)T^{2} \) |
| 11 | \( 1 + (-5.46 + 3.15i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.82 - 1.02i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-1.25 + 3.44i)T + (-13.0 - 10.9i)T^{2} \) |
| 19 | \( 1 + (-1.31 + 0.923i)T + (6.49 - 17.8i)T^{2} \) |
| 23 | \( 1 + (-1.45 + 2.51i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (8.93 + 2.39i)T + (25.1 + 14.5i)T^{2} \) |
| 31 | \( 1 + (-5.43 - 5.43i)T + 31iT^{2} \) |
| 41 | \( 1 + (-2.43 - 6.68i)T + (-31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 - 1.15T + 43T^{2} \) |
| 47 | \( 1 + (0.247 + 0.925i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-9.81 - 0.859i)T + (52.1 + 9.20i)T^{2} \) |
| 59 | \( 1 + (-4.09 - 0.357i)T + (58.1 + 10.2i)T^{2} \) |
| 61 | \( 1 + (8.91 + 4.15i)T + (39.2 + 46.7i)T^{2} \) |
| 67 | \( 1 + (-1.01 - 11.5i)T + (-65.9 + 11.6i)T^{2} \) |
| 71 | \( 1 + (-0.482 + 2.73i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (0.427 - 0.427i)T - 73iT^{2} \) |
| 79 | \( 1 + (-0.0358 - 0.410i)T + (-77.7 + 13.7i)T^{2} \) |
| 83 | \( 1 + (3.12 + 6.71i)T + (-53.3 + 63.5i)T^{2} \) |
| 89 | \( 1 + (0.989 - 11.3i)T + (-87.6 - 15.4i)T^{2} \) |
| 97 | \( 1 + (-4.75 - 2.74i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.68548210207652260380877995474, −9.804679904983708222013146562207, −9.290814454255228720148171689818, −8.783573818949103197118548545880, −7.21302811750891292761060850201, −6.48198229903264080719721933555, −5.64996478273763110637821924827, −4.25274779387213281176410426521, −3.31282728450000762301085027955, −0.72051874051535926135590784732,
1.98844970015779882852694765029, 3.31898154066969400072937339906, 4.16575606374004575688617061835, 5.76339484862569998526161114641, 6.82734765498105915281480824623, 7.54495816828494291066166828382, 9.244734648771919092187210079918, 9.905628985948806235465108070231, 10.42506478172213035025931861153, 11.55462299191347690637439026505
