| L(s) = 1 | + (−9.23 + 15.9i)2-s + (−68.0 − 117. i)3-s + (85.4 + 147. i)4-s + (−1.30e3 − 2.26e3i)5-s + 2.51e3·6-s + (−1.26e3 − 2.18e3i)7-s − 1.26e4·8-s + (571. − 989. i)9-s + 4.82e4·10-s + 1.08e4·11-s + (1.16e4 − 2.01e4i)12-s + (1.13e3 + 1.96e3i)13-s + 4.66e4·14-s + (−1.77e5 + 3.07e5i)15-s + (7.27e4 − 1.25e5i)16-s + (1.68e4 − 2.91e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.408 + 0.706i)2-s + (−0.485 − 0.840i)3-s + (0.166 + 0.289i)4-s + (−0.934 − 1.61i)5-s + 0.792·6-s + (−0.198 − 0.344i)7-s − 1.08·8-s + (0.0290 − 0.0502i)9-s + 1.52·10-s + 0.222·11-s + (0.161 − 0.280i)12-s + (0.0109 + 0.0190i)13-s + 0.324·14-s + (−0.906 + 1.57i)15-s + (0.277 − 0.480i)16-s + (0.0489 − 0.0847i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.644 - 0.764i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.644 - 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.0392610 + 0.0844913i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0392610 + 0.0844913i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (5.40e6 - 1.00e7i)T \) |
| good | 2 | \( 1 + (9.23 - 15.9i)T + (-256 - 443. i)T^{2} \) |
| 3 | \( 1 + (68.0 + 117. i)T + (-9.84e3 + 1.70e4i)T^{2} \) |
| 5 | \( 1 + (1.30e3 + 2.26e3i)T + (-9.76e5 + 1.69e6i)T^{2} \) |
| 7 | \( 1 + (1.26e3 + 2.18e3i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 - 1.08e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + (-1.13e3 - 1.96e3i)T + (-5.30e9 + 9.18e9i)T^{2} \) |
| 17 | \( 1 + (-1.68e4 + 2.91e4i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 19 | \( 1 + (-2.40e5 - 4.16e5i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + 7.89e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 5.28e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 3.80e6T + 2.64e13T^{2} \) |
| 41 | \( 1 + (-1.09e7 - 1.90e7i)T + (-1.63e14 + 2.83e14i)T^{2} \) |
| 43 | \( 1 - 2.77e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.35e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + (-2.20e7 + 3.82e7i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (7.48e7 - 1.29e8i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (-1.96e7 - 3.40e7i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-3.65e7 - 6.32e7i)T + (-1.36e16 + 2.35e16i)T^{2} \) |
| 71 | \( 1 + (1.53e8 + 2.66e8i)T + (-2.29e16 + 3.97e16i)T^{2} \) |
| 73 | \( 1 + 2.90e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + (-2.50e8 - 4.34e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + (-1.57e8 + 2.73e8i)T + (-9.34e16 - 1.61e17i)T^{2} \) |
| 89 | \( 1 + (1.12e8 - 1.94e8i)T + (-1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 + 8.36e8T + 7.60e17T^{2} \) |
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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
−15.24912794390848869181643743039, −13.24536745298205585880285447336, −12.26127310125171657739995142585, −11.73948305889166508188706966531, −9.314570418325013729113469367661, −8.103801189627853763538579132954, −7.27509114566838672479768345923, −5.80828631483468905654629107262, −3.94533650422564748122827747030, −1.13241717718446491801264990127,
0.04790287783989021386321762953, 2.49237429124854029494798788773, 3.84629365684922424716627761745, 5.89640789431106152900894371534, 7.35301934927845377253003784225, 9.402919966989815826713047458743, 10.58234530029911357765203570363, 11.07658513377306223557199404774, 12.04939102193924897296712920096, 14.27889965175977755979996913618
