| L(s) = 1 | + (−1.04 + 1.24i)2-s + (0.406 + 1.11i)3-s + (−0.107 − 0.611i)4-s + (2.23 + 0.0626i)5-s + (−1.80 − 0.658i)6-s + (−1.93 + 1.11i)7-s + (−1.93 − 1.11i)8-s + (1.21 − 1.02i)9-s + (−2.40 + 2.70i)10-s + (−2.82 + 4.88i)11-s + (0.639 − 0.369i)12-s + (1.60 − 4.41i)13-s + (0.627 − 3.55i)14-s + (0.838 + 2.52i)15-s + (4.56 − 1.66i)16-s + (−0.505 + 0.601i)17-s + ⋯ |
| L(s) = 1 | + (−0.735 + 0.877i)2-s + (0.234 + 0.644i)3-s + (−0.0539 − 0.305i)4-s + (0.999 + 0.0280i)5-s + (−0.738 − 0.268i)6-s + (−0.730 + 0.421i)7-s + (−0.683 − 0.394i)8-s + (0.405 − 0.340i)9-s + (−0.760 + 0.856i)10-s + (−0.850 + 1.47i)11-s + (0.184 − 0.106i)12-s + (0.445 − 1.22i)13-s + (0.167 − 0.951i)14-s + (0.216 + 0.651i)15-s + (1.14 − 0.415i)16-s + (−0.122 + 0.146i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.376 - 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.376 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.449809 + 0.668729i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.449809 + 0.668729i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-2.23 - 0.0626i)T \) |
| 19 | \( 1 + (-2.63 + 3.47i)T \) |
| good | 2 | \( 1 + (1.04 - 1.24i)T + (-0.347 - 1.96i)T^{2} \) |
| 3 | \( 1 + (-0.406 - 1.11i)T + (-2.29 + 1.92i)T^{2} \) |
| 7 | \( 1 + (1.93 - 1.11i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.82 - 4.88i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.60 + 4.41i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (0.505 - 0.601i)T + (-2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (-5.05 + 0.890i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (2.32 - 1.95i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (4.05 + 7.01i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 0.985iT - 37T^{2} \) |
| 41 | \( 1 + (1.33 - 0.484i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (1.49 + 0.264i)T + (40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (0.617 + 0.735i)T + (-8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (4.34 - 0.766i)T + (49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (7.56 + 6.34i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-0.363 - 2.06i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-2.08 - 2.48i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.24 + 7.06i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (5.18 + 14.2i)T + (-55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-1.25 + 0.458i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (6.51 - 3.76i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-3.19 - 1.16i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (8.20 - 9.77i)T + (-16.8 - 95.5i)T^{2} \) |
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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
−15.02484559838650853884869416049, −13.08532617040441802490897984094, −12.64227625333068142772192474508, −10.52279649430621930515321001486, −9.608703175045962651706535877612, −9.126182703945787992734806330096, −7.58701016749982985447898588862, −6.49386050629935759469487952037, −5.18751791689828233068295966235, −3.02749093397213183112343749624,
1.49144560318358168215574643695, 3.04427075003967797839706645314, 5.62869195217203849608598416904, 6.86249758912112764098240590195, 8.459895520081180890552402822810, 9.439579678560988815170663640667, 10.41292064925123861246315128657, 11.23789273840867837606592905758, 12.69892108107450902626946542832, 13.53515916469772086748937050214
