| L(s) = 1 | + (−0.480 + 0.832i)2-s + (12.5 + 21.7i)3-s + (15.5 + 26.9i)4-s + (14.2 + 24.6i)5-s − 24.1·6-s + (−96.5 − 167. i)7-s − 60.6·8-s + (−193. + 335. i)9-s − 27.3·10-s + 478.·11-s + (−390. + 675. i)12-s + (−111. − 192. i)13-s + 185.·14-s + (−357. + 619. i)15-s + (−468. + 810. i)16-s + (447. − 775. i)17-s + ⋯ |
| L(s) = 1 | + (−0.0849 + 0.147i)2-s + (0.805 + 1.39i)3-s + (0.485 + 0.841i)4-s + (0.254 + 0.441i)5-s − 0.273·6-s + (−0.744 − 1.29i)7-s − 0.334·8-s + (−0.797 + 1.38i)9-s − 0.0866·10-s + 1.19·11-s + (−0.782 + 1.35i)12-s + (−0.182 − 0.316i)13-s + 0.253·14-s + (−0.410 + 0.710i)15-s + (−0.457 + 0.791i)16-s + (0.375 − 0.650i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.388 - 0.921i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.388 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.13644 + 1.71300i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.13644 + 1.71300i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (5.94e3 - 5.83e3i)T \) |
| good | 2 | \( 1 + (0.480 - 0.832i)T + (-16 - 27.7i)T^{2} \) |
| 3 | \( 1 + (-12.5 - 21.7i)T + (-121.5 + 210. i)T^{2} \) |
| 5 | \( 1 + (-14.2 - 24.6i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 7 | \( 1 + (96.5 + 167. i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 - 478.T + 1.61e5T^{2} \) |
| 13 | \( 1 + (111. + 192. i)T + (-1.85e5 + 3.21e5i)T^{2} \) |
| 17 | \( 1 + (-447. + 775. i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-1.00e3 - 1.74e3i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + 2.89e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.82e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.93e3T + 2.86e7T^{2} \) |
| 41 | \( 1 + (1.59e3 + 2.76e3i)T + (-5.79e7 + 1.00e8i)T^{2} \) |
| 43 | \( 1 - 1.90e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.58e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + (-1.39e4 + 2.41e4i)T + (-2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + (-1.28e4 + 2.21e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-5.72e3 - 9.90e3i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (4.82e3 + 8.36e3i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + (1.75e4 + 3.03e4i)T + (-9.02e8 + 1.56e9i)T^{2} \) |
| 73 | \( 1 - 3.99e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + (3.25e4 + 5.63e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-5.65e3 + 9.80e3i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 + (5.31e4 - 9.20e4i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 - 7.97e4T + 8.58e9T^{2} \) |
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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
−16.03397869097998251239167863499, −14.53901874462197577140507244667, −13.78857556484194861169109886154, −11.98685594939267282107093489323, −10.38651664095541621818631846999, −9.666569164430574042790953066495, −8.116585907714335978568819598240, −6.67122510265840655657839813641, −4.04818224724072915340974730885, −3.15327846627316421437592083297,
1.30493798374440984877377866653, 2.61891687421467904953661419798, 5.89649164197559826160819169684, 6.88805717449843450927621264887, 8.726330848024209195041945398672, 9.564255729630501492265250015535, 11.76665128259323113035320692616, 12.50617503548367272524371786398, 13.81091244628994248942131122878, 14.76744679680486132812429232093
