| L(s) = 1 | + (0.806 − 0.465i)2-s + (−38.5 + 66.7i)3-s + (−63.5 + 110. i)4-s + (−299. − 172. i)5-s + 71.8i·6-s + (−372. + 645. i)7-s + 237. i·8-s + (−1.87e3 − 3.25e3i)9-s − 321.·10-s + 7.38e3·11-s + (−4.90e3 − 8.48e3i)12-s + (1.00e4 + 5.78e3i)13-s + 694. i·14-s + (2.30e4 − 1.33e4i)15-s + (−8.02e3 − 1.39e4i)16-s + (−2.41e4 + 1.39e4i)17-s + ⋯ |
| L(s) = 1 | + (0.0713 − 0.0411i)2-s + (−0.824 + 1.42i)3-s + (−0.496 + 0.860i)4-s + (−1.07 − 0.617i)5-s + 0.135i·6-s + (−0.410 + 0.711i)7-s + 0.164i·8-s + (−0.859 − 1.48i)9-s − 0.101·10-s + 1.67·11-s + (−0.818 − 1.41i)12-s + (1.26 + 0.729i)13-s + 0.0676i·14-s + (1.76 − 1.01i)15-s + (−0.489 − 0.848i)16-s + (−1.19 + 0.688i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0708 + 0.997i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.0708 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0900138 - 0.0838453i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0900138 - 0.0838453i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (-2.94e5 + 9.11e4i)T \) |
| good | 2 | \( 1 + (-0.806 + 0.465i)T + (64 - 110. i)T^{2} \) |
| 3 | \( 1 + (38.5 - 66.7i)T + (-1.09e3 - 1.89e3i)T^{2} \) |
| 5 | \( 1 + (299. + 172. i)T + (3.90e4 + 6.76e4i)T^{2} \) |
| 7 | \( 1 + (372. - 645. i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 - 7.38e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (-1.00e4 - 5.78e3i)T + (3.13e7 + 5.43e7i)T^{2} \) |
| 17 | \( 1 + (2.41e4 - 1.39e4i)T + (2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (4.27e4 + 2.46e4i)T + (4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 - 5.29e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 - 1.22e3iT - 1.72e10T^{2} \) |
| 31 | \( 1 + 2.31e5iT - 2.75e10T^{2} \) |
| 41 | \( 1 + (1.21e5 - 2.10e5i)T + (-9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + 4.56e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 + 3.32e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + (-2.61e5 - 4.52e5i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + (-1.06e6 + 6.16e5i)T + (1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (5.49e5 + 3.17e5i)T + (1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (1.49e6 - 2.59e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-3.34e5 + 5.79e5i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 + 2.98e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + (2.32e6 + 1.34e6i)T + (9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (3.23e6 + 5.59e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (-1.11e6 + 6.43e5i)T + (2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 - 1.61e6iT - 8.07e13T^{2} \) |
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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
−15.89381979145698107075553422353, −15.05978095157482001701594741556, −13.10132276216889498777747869544, −11.75292558383858970015653058748, −11.29906881314388576662649249861, −9.169897828964020190836100794411, −8.722463786604742914515304869319, −6.27872597660390678883721175282, −4.22932122254015971010110884859, −4.01318693692000522648494137566,
0.07055377111018688842387216873, 1.21721034510500857205184909098, 4.07171510776474949012858658428, 6.30771860476637154198411972107, 6.81942211666709561854104790922, 8.518843047456537971740861755769, 10.64229861404593619893211479711, 11.45494908452224876049439414491, 12.76964815737486747185317314980, 13.79969252589444597131877326340
