| L(s) = 1 | + (1.61 + 1.17i)2-s + (−1.33 + 4.09i)3-s + (1.23 + 3.80i)4-s + (6.52 − 4.73i)5-s + (−6.97 + 5.06i)6-s + (−8.05 − 24.8i)7-s + (−2.47 + 7.60i)8-s + (6.82 + 4.95i)9-s + 16.1·10-s + (−33.3 − 14.7i)11-s − 17.2·12-s + (2.64 + 1.91i)13-s + (16.1 − 49.6i)14-s + (10.7 + 33.0i)15-s + (−12.9 + 9.40i)16-s + (16.8 − 12.2i)17-s + ⋯ |
| L(s) = 1 | + (0.572 + 0.415i)2-s + (−0.256 + 0.788i)3-s + (0.154 + 0.475i)4-s + (0.583 − 0.423i)5-s + (−0.474 + 0.344i)6-s + (−0.435 − 1.33i)7-s + (−0.109 + 0.336i)8-s + (0.252 + 0.183i)9-s + 0.509·10-s + (−0.914 − 0.403i)11-s − 0.414·12-s + (0.0563 + 0.0409i)13-s + (0.307 − 0.946i)14-s + (0.184 + 0.568i)15-s + (−0.202 + 0.146i)16-s + (0.240 − 0.175i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.648 - 0.760i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.648 - 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.25538 + 0.579360i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.25538 + 0.579360i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.61 - 1.17i)T \) |
| 11 | \( 1 + (33.3 + 14.7i)T \) |
| good | 3 | \( 1 + (1.33 - 4.09i)T + (-21.8 - 15.8i)T^{2} \) |
| 5 | \( 1 + (-6.52 + 4.73i)T + (38.6 - 118. i)T^{2} \) |
| 7 | \( 1 + (8.05 + 24.8i)T + (-277. + 201. i)T^{2} \) |
| 13 | \( 1 + (-2.64 - 1.91i)T + (678. + 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-16.8 + 12.2i)T + (1.51e3 - 4.67e3i)T^{2} \) |
| 19 | \( 1 + (38.9 - 119. i)T + (-5.54e3 - 4.03e3i)T^{2} \) |
| 23 | \( 1 - 97.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + (81.5 + 250. i)T + (-1.97e4 + 1.43e4i)T^{2} \) |
| 31 | \( 1 + (161. + 117. i)T + (9.20e3 + 2.83e4i)T^{2} \) |
| 37 | \( 1 + (-112. - 347. i)T + (-4.09e4 + 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-84.5 + 260. i)T + (-5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 - 388.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (16.0 - 49.3i)T + (-8.39e4 - 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-333. - 242. i)T + (4.60e4 + 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-8.12 - 24.9i)T + (-1.66e5 + 1.20e5i)T^{2} \) |
| 61 | \( 1 + (132. - 96.4i)T + (7.01e4 - 2.15e5i)T^{2} \) |
| 67 | \( 1 - 276.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-418. + 303. i)T + (1.10e5 - 3.40e5i)T^{2} \) |
| 73 | \( 1 + (74.6 + 229. i)T + (-3.14e5 + 2.28e5i)T^{2} \) |
| 79 | \( 1 + (-220. - 160. i)T + (1.52e5 + 4.68e5i)T^{2} \) |
| 83 | \( 1 + (58.7 - 42.6i)T + (1.76e5 - 5.43e5i)T^{2} \) |
| 89 | \( 1 + 1.19e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (1.18e3 + 860. i)T + (2.82e5 + 8.68e5i)T^{2} \) |
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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
−16.96342053102986890558572169470, −16.60211221328392733758517734881, −15.32525034441322959914110059819, −13.73531942292230278791621542805, −12.95196516929332497641643053131, −10.85807627699460035804845690535, −9.771042301484117589623539995578, −7.61592026690383926944713276178, −5.65424161763885758285245969751, −4.06093736295486618416107366091,
2.46453744128779366832478711766, 5.52804045030340407986287728335, 6.90775298407155775345447574611, 9.280129257696872031992573259402, 10.91738940241206495474703035528, 12.53293475086121485769664569538, 13.01667032689774305635431671870, 14.71039489005093166359882320151, 15.81154759608634443745844081927, 17.91478948919482668202204300445
