| L(s) = 1 | + (−3.85 − 0.680i)2-s + (0.216 + 1.22i)3-s + (6.89 + 2.51i)4-s + (1.80 + 2.15i)5-s − 4.87i·6-s + (24.0 − 20.2i)7-s + (2.24 + 1.29i)8-s + (23.9 − 8.70i)9-s + (−5.50 − 9.53i)10-s + (−4.91 + 8.51i)11-s + (−1.58 + 8.99i)12-s + (−3.87 + 10.6i)13-s + (−106. + 61.5i)14-s + (−2.24 + 2.68i)15-s + (−52.7 − 44.2i)16-s + (13.4 + 37.0i)17-s + ⋯ |
| L(s) = 1 | + (−1.36 − 0.240i)2-s + (0.0415 + 0.235i)3-s + (0.862 + 0.313i)4-s + (0.161 + 0.192i)5-s − 0.331i·6-s + (1.30 − 1.09i)7-s + (0.0990 + 0.0571i)8-s + (0.885 − 0.322i)9-s + (−0.174 − 0.301i)10-s + (−0.134 + 0.233i)11-s + (−0.0381 + 0.216i)12-s + (−0.0827 + 0.227i)13-s + (−2.03 + 1.17i)14-s + (−0.0387 + 0.0461i)15-s + (−0.824 − 0.691i)16-s + (0.192 + 0.528i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.368i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.772743 - 0.147410i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.772743 - 0.147410i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (-50.8 - 219. i)T \) |
| good | 2 | \( 1 + (3.85 + 0.680i)T + (7.51 + 2.73i)T^{2} \) |
| 3 | \( 1 + (-0.216 - 1.22i)T + (-25.3 + 9.23i)T^{2} \) |
| 5 | \( 1 + (-1.80 - 2.15i)T + (-21.7 + 123. i)T^{2} \) |
| 7 | \( 1 + (-24.0 + 20.2i)T + (59.5 - 337. i)T^{2} \) |
| 11 | \( 1 + (4.91 - 8.51i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (3.87 - 10.6i)T + (-1.68e3 - 1.41e3i)T^{2} \) |
| 17 | \( 1 + (-13.4 - 37.0i)T + (-3.76e3 + 3.15e3i)T^{2} \) |
| 19 | \( 1 + (-70.8 + 12.4i)T + (6.44e3 - 2.34e3i)T^{2} \) |
| 23 | \( 1 + (31.6 - 18.2i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (62.6 + 36.1i)T + (1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + 102. iT - 2.97e4T^{2} \) |
| 41 | \( 1 + (483. + 175. i)T + (5.27e4 + 4.43e4i)T^{2} \) |
| 43 | \( 1 - 355. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (-166. - 288. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (290. + 244. i)T + (2.58e4 + 1.46e5i)T^{2} \) |
| 59 | \( 1 + (-269. + 321. i)T + (-3.56e4 - 2.02e5i)T^{2} \) |
| 61 | \( 1 + (263. - 722. i)T + (-1.73e5 - 1.45e5i)T^{2} \) |
| 67 | \( 1 + (377. - 316. i)T + (5.22e4 - 2.96e5i)T^{2} \) |
| 71 | \( 1 + (-124. - 708. i)T + (-3.36e5 + 1.22e5i)T^{2} \) |
| 73 | \( 1 + 492.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (644. + 767. i)T + (-8.56e4 + 4.85e5i)T^{2} \) |
| 83 | \( 1 + (-909. + 330. i)T + (4.38e5 - 3.67e5i)T^{2} \) |
| 89 | \( 1 + (326. - 388. i)T + (-1.22e5 - 6.94e5i)T^{2} \) |
| 97 | \( 1 + (561. - 323. i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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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
−16.21895570193172426998317533585, −14.74587216203873140564402790379, −13.56899680767952327107521634343, −11.64824205767083536470291676377, −10.50586236611568073533972427316, −9.756826321224364456539168479950, −8.175492123002410449665360910685, −7.18455161835433330085039622136, −4.50475006676652715294202722120, −1.40707235109880147922150477444,
1.64729649815148832143566379805, 5.18449590542771865424076039269, 7.32495077491907109545124923648, 8.326740210214075120253532751031, 9.420636686887679481600068549650, 10.77472254218589953974925834724, 12.09920044728593518798032964634, 13.63824543508129276197244425748, 15.17170036082266117972865191643, 16.17717970663802504957628388257
