| L(s) = 1 | + (−3.52 − 4.19i)2-s + (5.42 + 4.55i)3-s + (−3.82 + 21.7i)4-s + (4.86 + 13.3i)5-s − 38.8i·6-s + (−13.3 + 4.86i)7-s + (66.6 − 38.4i)8-s + (4.02 + 22.8i)9-s + (39.0 − 67.5i)10-s + (2.49 + 4.32i)11-s + (−119. + 100. i)12-s + (−12.4 − 2.18i)13-s + (67.5 + 38.9i)14-s + (−34.4 + 94.7i)15-s + (−230. − 83.9i)16-s + (96.2 − 16.9i)17-s + ⋯ |
| L(s) = 1 | + (−1.24 − 1.48i)2-s + (1.04 + 0.876i)3-s + (−0.478 + 2.71i)4-s + (0.435 + 1.19i)5-s − 2.64i·6-s + (−0.721 + 0.262i)7-s + (2.94 − 1.70i)8-s + (0.149 + 0.845i)9-s + (1.23 − 2.13i)10-s + (0.0683 + 0.118i)11-s + (−2.87 + 2.41i)12-s + (−0.264 − 0.0466i)13-s + (1.28 + 0.744i)14-s + (−0.593 + 1.63i)15-s + (−3.60 − 1.31i)16-s + (1.37 − 0.242i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.203i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.979 - 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.911484 + 0.0935446i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.911484 + 0.0935446i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (112. - 194. i)T \) |
| good | 2 | \( 1 + (3.52 + 4.19i)T + (-1.38 + 7.87i)T^{2} \) |
| 3 | \( 1 + (-5.42 - 4.55i)T + (4.68 + 26.5i)T^{2} \) |
| 5 | \( 1 + (-4.86 - 13.3i)T + (-95.7 + 80.3i)T^{2} \) |
| 7 | \( 1 + (13.3 - 4.86i)T + (262. - 220. i)T^{2} \) |
| 11 | \( 1 + (-2.49 - 4.32i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (12.4 + 2.18i)T + (2.06e3 + 751. i)T^{2} \) |
| 17 | \( 1 + (-96.2 + 16.9i)T + (4.61e3 - 1.68e3i)T^{2} \) |
| 19 | \( 1 + (-28.6 + 34.1i)T + (-1.19e3 - 6.75e3i)T^{2} \) |
| 23 | \( 1 + (-46.0 - 26.5i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (85.0 - 49.1i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + 151. iT - 2.97e4T^{2} \) |
| 41 | \( 1 + (-66.6 + 377. i)T + (-6.47e4 - 2.35e4i)T^{2} \) |
| 43 | \( 1 - 141. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (-105. + 183. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-423. - 154. i)T + (1.14e5 + 9.56e4i)T^{2} \) |
| 59 | \( 1 + (-102. + 282. i)T + (-1.57e5 - 1.32e5i)T^{2} \) |
| 61 | \( 1 + (-688. - 121. i)T + (2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (885. - 322. i)T + (2.30e5 - 1.93e5i)T^{2} \) |
| 71 | \( 1 + (337. + 283. i)T + (6.21e4 + 3.52e5i)T^{2} \) |
| 73 | \( 1 + 118.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (331. + 910. i)T + (-3.77e5 + 3.16e5i)T^{2} \) |
| 83 | \( 1 + (124. + 706. i)T + (-5.37e5 + 1.95e5i)T^{2} \) |
| 89 | \( 1 + (-146. + 403. i)T + (-5.40e5 - 4.53e5i)T^{2} \) |
| 97 | \( 1 + (-767. - 443. i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−16.14026437103403376299538693329, −14.69974945774825264389910804451, −13.39327236140215118183714954623, −11.88504736017028740170908341683, −10.44934996970277461463314544315, −9.829158609891864037499163040120, −8.977701861541258653756779349356, −7.41539564894063599074622918366, −3.50069834420907430325926130898, −2.65433116051449919069216325553,
1.21714118203406256523286251618, 5.54405825179016932232967549410, 7.07787695206777322621453437189, 8.143566092170976022934474830010, 9.054404017062951507855505882443, 10.00115131336077717750947181539, 12.77886560047862663803376237187, 13.84783774656444533915460217147, 14.75709645977569134302961641773, 16.29374128370690213444188889001
