| L(s) = 1 | + (0.399 − 1.09i)2-s + (0.610 − 0.222i)3-s + (5.08 + 4.26i)4-s + (6.35 + 1.12i)5-s − 0.760i·6-s + (4.53 − 25.7i)7-s + (14.8 − 8.55i)8-s + (−20.3 + 17.0i)9-s + (3.77 − 6.53i)10-s + (6.67 + 11.5i)11-s + (4.05 + 1.47i)12-s + (−36.5 + 43.5i)13-s + (−26.4 − 15.2i)14-s + (4.13 − 0.728i)15-s + (5.73 + 32.5i)16-s + (−77.5 − 92.3i)17-s + ⋯ |
| L(s) = 1 | + (0.141 − 0.388i)2-s + (0.117 − 0.0427i)3-s + (0.635 + 0.532i)4-s + (0.568 + 0.100i)5-s − 0.0517i·6-s + (0.244 − 1.38i)7-s + (0.654 − 0.378i)8-s + (−0.754 + 0.632i)9-s + (0.119 − 0.206i)10-s + (0.182 + 0.316i)11-s + (0.0974 + 0.0354i)12-s + (−0.778 + 0.928i)13-s + (−0.504 − 0.291i)14-s + (0.0711 − 0.0125i)15-s + (0.0896 + 0.508i)16-s + (−1.10 − 1.31i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 + 0.332i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.943 + 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.60638 - 0.274446i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.60638 - 0.274446i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (-172. + 145. i)T \) |
| good | 2 | \( 1 + (-0.399 + 1.09i)T + (-6.12 - 5.14i)T^{2} \) |
| 3 | \( 1 + (-0.610 + 0.222i)T + (20.6 - 17.3i)T^{2} \) |
| 5 | \( 1 + (-6.35 - 1.12i)T + (117. + 42.7i)T^{2} \) |
| 7 | \( 1 + (-4.53 + 25.7i)T + (-322. - 117. i)T^{2} \) |
| 11 | \( 1 + (-6.67 - 11.5i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (36.5 - 43.5i)T + (-381. - 2.16e3i)T^{2} \) |
| 17 | \( 1 + (77.5 + 92.3i)T + (-853. + 4.83e3i)T^{2} \) |
| 19 | \( 1 + (-26.7 - 73.4i)T + (-5.25e3 + 4.40e3i)T^{2} \) |
| 23 | \( 1 + (105. + 60.7i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-168. + 97.5i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 - 132. iT - 2.97e4T^{2} \) |
| 41 | \( 1 + (-1.20 - 1.00i)T + (1.19e4 + 6.78e4i)T^{2} \) |
| 43 | \( 1 - 306. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (-114. + 197. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-7.64 - 43.3i)T + (-1.39e5 + 5.09e4i)T^{2} \) |
| 59 | \( 1 + (183. - 32.3i)T + (1.92e5 - 7.02e4i)T^{2} \) |
| 61 | \( 1 + (-38.7 + 46.1i)T + (-3.94e4 - 2.23e5i)T^{2} \) |
| 67 | \( 1 + (65.2 - 369. i)T + (-2.82e5 - 1.02e5i)T^{2} \) |
| 71 | \( 1 + (-962. + 350. i)T + (2.74e5 - 2.30e5i)T^{2} \) |
| 73 | \( 1 - 920.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-15.1 - 2.66i)T + (4.63e5 + 1.68e5i)T^{2} \) |
| 83 | \( 1 + (-335. + 281. i)T + (9.92e4 - 5.63e5i)T^{2} \) |
| 89 | \( 1 + (783. - 138. i)T + (6.62e5 - 2.41e5i)T^{2} \) |
| 97 | \( 1 + (1.48e3 + 859. i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−16.15633196028117601031347646655, −14.15410835519855478852537210164, −13.68960073890583954490303343784, −12.06008360160056496282848585403, −11.01662678957785727129925408668, −9.847353624550636748105208179147, −7.87824001348961372569401668817, −6.73324329718374732202770044531, −4.37969563382782560944598547841, −2.28824433272678904371815432740,
2.39918414365912197948963419171, 5.43779066914911178911845681050, 6.28902817740764585529073805737, 8.292442899565343526231930574287, 9.597254984916669023786856014096, 11.14120966584643197466178882256, 12.27609928745911153496131746072, 13.86288217480408600440210224221, 15.12762125270338667222527873557, 15.45716996555491435485396573485
