| L(s) = 1 | − 22.3·2-s + (16.6 + 28.9i)3-s + 369.·4-s + (−15.2 + 26.3i)5-s + (−372. − 645. i)6-s + (−91.5 − 158. i)7-s − 5.39e3·8-s + (535. − 928. i)9-s + (339. − 587. i)10-s + (−515. + 892. i)11-s + (6.17e3 + 1.06e4i)12-s + (5.01e3 − 8.69e3i)13-s + (2.04e3 + 3.53e3i)14-s − 1.01e3·15-s + 7.29e4·16-s + (1.47e4 + 2.56e4i)17-s + ⋯ |
| L(s) = 1 | − 1.97·2-s + (0.357 + 0.618i)3-s + 2.88·4-s + (−0.0544 + 0.0942i)5-s + (−0.704 − 1.21i)6-s + (−0.100 − 0.174i)7-s − 3.72·8-s + (0.245 − 0.424i)9-s + (0.107 − 0.185i)10-s + (−0.116 + 0.202i)11-s + (1.03 + 1.78i)12-s + (0.633 − 1.09i)13-s + (0.198 + 0.344i)14-s − 0.0777·15-s + 4.45·16-s + (0.730 + 1.26i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 - 0.323i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.946 - 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.814529 + 0.135290i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.814529 + 0.135290i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 + (-1.65e5 - 1.19e4i)T \) |
| good | 2 | \( 1 + 22.3T + 128T^{2} \) |
| 3 | \( 1 + (-16.6 - 28.9i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + (15.2 - 26.3i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 7 | \( 1 + (91.5 + 158. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (515. - 892. i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + (-5.01e3 + 8.69e3i)T + (-3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 + (-1.47e4 - 2.56e4i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (1.48e4 + 2.57e4i)T + (-4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 - 1.49e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 5.60e4T + 1.72e10T^{2} \) |
| 37 | \( 1 + (-2.24e5 - 3.89e5i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 + (-2.85e5 + 4.95e5i)T + (-9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (-1.94e5 - 3.37e5i)T + (-1.35e11 + 2.35e11i)T^{2} \) |
| 47 | \( 1 + 1.90e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + (8.97e5 - 1.55e6i)T + (-5.87e11 - 1.01e12i)T^{2} \) |
| 59 | \( 1 + (-3.98e4 - 6.89e4i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + 1.43e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + (-1.92e6 + 3.33e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-1.54e6 + 2.68e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 + (9.15e5 - 1.58e6i)T + (-5.52e12 - 9.56e12i)T^{2} \) |
| 79 | \( 1 + (-8.53e5 - 1.47e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (-2.93e6 + 5.08e6i)T + (-1.35e13 - 2.35e13i)T^{2} \) |
| 89 | \( 1 - 4.57e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.24e7T + 8.07e13T^{2} \) |
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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
−15.52962489576563087690967339890, −15.09550949061132893130269632189, −12.51957233880048570218072649400, −10.87929822775802469726886547581, −10.12663204539653491227892577749, −8.974374278690065177172290138315, −7.894890936121656539837631848717, −6.38248039131823665497390421954, −3.11596723467478966445321441189, −0.991891104340839940539826711604,
1.02843551263729391336483263294, 2.47188830165375009914195099024, 6.44243428020262223076836131970, 7.65551493314315359936118885860, 8.643182317589139694419552849420, 9.860155437618381617726882510652, 11.17015738255575718721418962259, 12.36659682694592253809194126043, 14.27419460518557436858107793819, 16.03404245581350247943187980041
