| 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
−16.17717970663802504957628388257, −15.17170036082266117972865191643, −13.63824543508129276197244425748, −12.09920044728593518798032964634, −10.77472254218589953974925834724, −9.420636686887679481600068549650, −8.326740210214075120253532751031, −7.32495077491907109545124923648, −5.18449590542771865424076039269, −1.64729649815148832143566379805,
1.40707235109880147922150477444, 4.50475006676652715294202722120, 7.18455161835433330085039622136, 8.175492123002410449665360910685, 9.756826321224364456539168479950, 10.50586236611568073533972427316, 11.64824205767083536470291676377, 13.56899680767952327107521634343, 14.74587216203873140564402790379, 16.21895570193172426998317533585
