| L(s) = 1 | + (0.841 − 0.540i)2-s + (−0.311 + 2.16i)3-s + (0.415 − 0.909i)4-s + (3.63 − 1.06i)5-s + (0.909 + 1.99i)6-s + (0.654 + 0.755i)7-s + (−0.142 − 0.989i)8-s + (−1.71 − 0.504i)9-s + (2.48 − 2.86i)10-s + (−4.18 − 2.69i)11-s + (1.84 + 1.18i)12-s + (−0.479 + 0.553i)13-s + (0.959 + 0.281i)14-s + (1.17 + 8.20i)15-s + (−0.654 − 0.755i)16-s + (1.29 + 2.82i)17-s + ⋯ |
| L(s) = 1 | + (0.594 − 0.382i)2-s + (−0.179 + 1.25i)3-s + (0.207 − 0.454i)4-s + (1.62 − 0.477i)5-s + (0.371 + 0.812i)6-s + (0.247 + 0.285i)7-s + (−0.0503 − 0.349i)8-s + (−0.573 − 0.168i)9-s + (0.784 − 0.905i)10-s + (−1.26 − 0.811i)11-s + (0.531 + 0.341i)12-s + (−0.132 + 0.153i)13-s + (0.256 + 0.0752i)14-s + (0.304 + 2.11i)15-s + (−0.163 − 0.188i)16-s + (0.313 + 0.686i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.249i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 - 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.07750 + 0.263532i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.07750 + 0.263532i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.841 + 0.540i)T \) |
| 7 | \( 1 + (-0.654 - 0.755i)T \) |
| 23 | \( 1 + (0.648 - 4.75i)T \) |
| good | 3 | \( 1 + (0.311 - 2.16i)T + (-2.87 - 0.845i)T^{2} \) |
| 5 | \( 1 + (-3.63 + 1.06i)T + (4.20 - 2.70i)T^{2} \) |
| 11 | \( 1 + (4.18 + 2.69i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (0.479 - 0.553i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.29 - 2.82i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-0.406 + 0.890i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (0.0957 + 0.209i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (1.20 + 8.41i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (5.60 + 1.64i)T + (31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (-0.499 + 0.146i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (1.47 - 10.2i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 + 11.5T + 47T^{2} \) |
| 53 | \( 1 + (1.08 + 1.25i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-7.56 + 8.73i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (1.65 + 11.5i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (3.56 - 2.29i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (-3.45 + 2.21i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-3.34 + 7.32i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (-2.51 + 2.90i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (5.41 + 1.59i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (2.52 - 17.5i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-7.31 + 2.14i)T + (81.6 - 52.4i)T^{2} \) |
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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
−11.40171493610638878626002364041, −10.67663899276205820890343841135, −9.838721412208118982007184499171, −9.384122746980765107773785091856, −8.100182097759248102434348306242, −6.16270630260500879561727073560, −5.41104670833952204001159667250, −4.85758496861839516924559046268, −3.36846406719437124908768040769, −1.96390554674323635700478484611,
1.80307079962809888516330579745, 2.78139449417463930768080189369, 4.96767507082717702057463952753, 5.73872299800635618777035464716, 6.85141210634413724633121307585, 7.25672962299105414821506838746, 8.476294559682566230032247363939, 9.982660945874659327781901711434, 10.55196482796965429195267187434, 11.98402817002558543986513232297
