| L(s) = 1 | + (−3.29 + 1.20i)3-s + (−0.0631 + 0.357i)5-s + (−13.5 − 23.4i)7-s + (−11.2 + 9.43i)9-s + (28.4 − 49.2i)11-s + (−72.8 − 26.5i)13-s + (−0.221 − 1.25i)15-s + (−57.2 − 48.0i)17-s + (28.0 + 77.9i)19-s + (72.9 + 61.2i)21-s + (−1.58 − 8.99i)23-s + (117. + 42.7i)25-s + (73.1 − 126. i)27-s + (−165. + 139. i)29-s + (31.6 + 54.8i)31-s + ⋯ |
| L(s) = 1 | + (−0.634 + 0.231i)3-s + (−0.00564 + 0.0320i)5-s + (−0.732 − 1.26i)7-s + (−0.416 + 0.349i)9-s + (0.779 − 1.35i)11-s + (−1.55 − 0.565i)13-s + (−0.00381 − 0.0216i)15-s + (−0.816 − 0.685i)17-s + (0.339 + 0.940i)19-s + (0.758 + 0.636i)21-s + (−0.0143 − 0.0815i)23-s + (0.938 + 0.341i)25-s + (0.521 − 0.903i)27-s + (−1.06 + 0.891i)29-s + (0.183 + 0.317i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.633 + 0.773i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.633 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.228975 - 0.483364i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.228975 - 0.483364i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-28.0 - 77.9i)T \) |
| good | 3 | \( 1 + (3.29 - 1.20i)T + (20.6 - 17.3i)T^{2} \) |
| 5 | \( 1 + (0.0631 - 0.357i)T + (-117. - 42.7i)T^{2} \) |
| 7 | \( 1 + (13.5 + 23.4i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (-28.4 + 49.2i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (72.8 + 26.5i)T + (1.68e3 + 1.41e3i)T^{2} \) |
| 17 | \( 1 + (57.2 + 48.0i)T + (853. + 4.83e3i)T^{2} \) |
| 23 | \( 1 + (1.58 + 8.99i)T + (-1.14e4 + 4.16e3i)T^{2} \) |
| 29 | \( 1 + (165. - 139. i)T + (4.23e3 - 2.40e4i)T^{2} \) |
| 31 | \( 1 + (-31.6 - 54.8i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 - 109.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-55.2 + 20.0i)T + (5.27e4 - 4.43e4i)T^{2} \) |
| 43 | \( 1 + (-58.1 + 329. i)T + (-7.47e4 - 2.71e4i)T^{2} \) |
| 47 | \( 1 + (266. - 223. i)T + (1.80e4 - 1.02e5i)T^{2} \) |
| 53 | \( 1 + (-30.0 - 170. i)T + (-1.39e5 + 5.09e4i)T^{2} \) |
| 59 | \( 1 + (647. + 543. i)T + (3.56e4 + 2.02e5i)T^{2} \) |
| 61 | \( 1 + (114. + 647. i)T + (-2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (-95.7 + 80.3i)T + (5.22e4 - 2.96e5i)T^{2} \) |
| 71 | \( 1 + (-65.1 + 369. i)T + (-3.36e5 - 1.22e5i)T^{2} \) |
| 73 | \( 1 + (-81.9 + 29.8i)T + (2.98e5 - 2.50e5i)T^{2} \) |
| 79 | \( 1 + (-512. + 186. i)T + (3.77e5 - 3.16e5i)T^{2} \) |
| 83 | \( 1 + (464. + 804. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-1.23e3 - 448. i)T + (5.40e5 + 4.53e5i)T^{2} \) |
| 97 | \( 1 + (-1.10e3 - 925. i)T + (1.58e5 + 8.98e5i)T^{2} \) |
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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
−13.71999902197045615808674136417, −12.48780023410992219333068580903, −11.22703721788929996712299481734, −10.46642167085601284735054934635, −9.253812525759328145480072755713, −7.61418183174060606226741836078, −6.39763810450052713296261319260, −4.99326983582783032791385827487, −3.32652369386048007431548183598, −0.34949978909090756841855170391,
2.43147978171155106403672103698, 4.68690156680169105493519896485, 6.13958963192407702145559041042, 7.08034874784945633130818228295, 9.027411746599025820695746011718, 9.689981727160731278171720369460, 11.48022525783916388990991515789, 12.18431653585339990846298283595, 12.92038847278132593298088171164, 14.75164733437219527497557162180
