| L(s) = 1 | + (0.295 + 1.38i)2-s + (−0.0218 − 1.73i)3-s + (−1.82 + 0.818i)4-s + (1.66 + 1.49i)5-s + (2.38 − 0.542i)6-s + 0.571·7-s + (−1.67 − 2.28i)8-s + (−2.99 + 0.0756i)9-s + (−1.57 + 2.74i)10-s + (5.16 + 3.75i)11-s + (1.45 + 3.14i)12-s + (1.72 + 2.37i)13-s + (0.169 + 0.790i)14-s + (2.54 − 2.91i)15-s + (2.66 − 2.98i)16-s + (1.33 − 4.10i)17-s + ⋯ |
| L(s) = 1 | + (0.209 + 0.977i)2-s + (−0.0126 − 0.999i)3-s + (−0.912 + 0.409i)4-s + (0.744 + 0.667i)5-s + (0.975 − 0.221i)6-s + 0.216·7-s + (−0.590 − 0.806i)8-s + (−0.999 + 0.0252i)9-s + (−0.497 + 0.867i)10-s + (1.55 + 1.13i)11-s + (0.420 + 0.907i)12-s + (0.478 + 0.659i)13-s + (0.0451 + 0.211i)14-s + (0.658 − 0.752i)15-s + (0.665 − 0.746i)16-s + (0.323 − 0.995i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.501 - 0.865i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.501 - 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.32067 + 0.761106i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.32067 + 0.761106i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.295 - 1.38i)T \) |
| 3 | \( 1 + (0.0218 + 1.73i)T \) |
| 5 | \( 1 + (-1.66 - 1.49i)T \) |
| good | 7 | \( 1 - 0.571T + 7T^{2} \) |
| 11 | \( 1 + (-5.16 - 3.75i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-1.72 - 2.37i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-1.33 + 4.10i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-1.41 - 0.460i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (0.0141 - 0.0195i)T + (-7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (1.13 - 0.369i)T + (23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (8.64 + 2.80i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (4.85 + 6.68i)T + (-11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-2.42 - 3.34i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 2.43T + 43T^{2} \) |
| 47 | \( 1 + (-3.48 + 1.13i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (1.79 + 5.53i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-4.98 + 3.62i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (8.41 + 6.11i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-3.09 + 9.53i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (1.01 + 3.13i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-4.98 + 6.86i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (5.80 - 1.88i)T + (63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-1.19 - 0.386i)T + (67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (8.73 - 12.0i)T + (-27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-9.99 + 3.24i)T + (78.4 - 57.0i)T^{2} \) |
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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
−12.10385145234736998403978187690, −11.21174088180799300184677765683, −9.531486922469615435740792827255, −9.101518422138471199508810326945, −7.62809056163628926966151800660, −6.92884250495440050134503092312, −6.30856262679019863563214538063, −5.18373606717561615442997623924, −3.61642165600224179287393348830, −1.78643351170511878669704436006,
1.35007183373554061938141640600, 3.26998953602516226631814326831, 4.14765830949020476256768922837, 5.42729945011619142660473360607, 6.04162597349302445765214676503, 8.518156071705618911759261920146, 8.882360601745284517365303289589, 9.830919913429881594953204235870, 10.67924440733399356875229862506, 11.43717533458299862598971617324
