| L(s) = 1 | + (−0.564 + 0.251i)2-s + (−0.978 + 0.207i)3-s + (−1.08 + 1.20i)4-s + (0.233 + 2.22i)5-s + (0.5 − 0.363i)6-s + (1.59 + 2.11i)7-s + (0.690 − 2.12i)8-s + (−1.82 + 0.813i)9-s + (−0.690 − 1.19i)10-s + (2.04 − 2.60i)11-s + (0.809 − 1.40i)12-s + (−0.690 − 0.502i)13-s + (−1.43 − 0.790i)14-s + (−0.690 − 2.12i)15-s + (−0.193 − 1.84i)16-s + (−1.12 − 0.502i)17-s + ⋯ |
| L(s) = 1 | + (−0.399 + 0.177i)2-s + (−0.564 + 0.120i)3-s + (−0.541 + 0.601i)4-s + (0.104 + 0.994i)5-s + (0.204 − 0.148i)6-s + (0.603 + 0.797i)7-s + (0.244 − 0.751i)8-s + (−0.609 + 0.271i)9-s + (−0.218 − 0.378i)10-s + (0.617 − 0.786i)11-s + (0.233 − 0.404i)12-s + (−0.191 − 0.139i)13-s + (−0.382 − 0.211i)14-s + (−0.178 − 0.549i)15-s + (−0.0484 − 0.460i)16-s + (−0.273 − 0.121i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0694 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0694 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.413630 + 0.443419i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.413630 + 0.443419i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-1.59 - 2.11i)T \) |
| 11 | \( 1 + (-2.04 + 2.60i)T \) |
| good | 2 | \( 1 + (0.564 - 0.251i)T + (1.33 - 1.48i)T^{2} \) |
| 3 | \( 1 + (0.978 - 0.207i)T + (2.74 - 1.22i)T^{2} \) |
| 5 | \( 1 + (-0.233 - 2.22i)T + (-4.89 + 1.03i)T^{2} \) |
| 13 | \( 1 + (0.690 + 0.502i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (1.12 + 0.502i)T + (11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (-5.25 - 5.83i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (-3.11 + 5.40i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.354 + 1.08i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (0.691 - 6.58i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (4.97 + 1.05i)T + (33.8 + 15.0i)T^{2} \) |
| 41 | \( 1 + (1.42 - 4.39i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 6.85T + 43T^{2} \) |
| 47 | \( 1 + (5.66 + 6.29i)T + (-4.91 + 46.7i)T^{2} \) |
| 53 | \( 1 + (-0.532 + 5.06i)T + (-51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (-2.48 + 2.75i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (0.725 + 6.90i)T + (-59.6 + 12.6i)T^{2} \) |
| 67 | \( 1 + (5.30 + 9.19i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (4.23 - 3.07i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-2.77 + 3.08i)T + (-7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (-0.646 + 0.288i)T + (52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (-7.28 + 5.29i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (5.23 - 9.06i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (2.30 + 1.67i)T + (29.9 + 92.2i)T^{2} \) |
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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
−14.57433483970111885202968491573, −13.98134496068663288278368025031, −12.35038090193135632408460533683, −11.47587315297298888115241319186, −10.41974807761003958908077253016, −8.976856340016117597389206383568, −8.027094011409560594288940648173, −6.54569970800395499160474917463, −5.16924825591601117619993026763, −3.18355983905936257973913749824,
1.12364637577154910260252882813, 4.55355169743199541460354082713, 5.46576790372544036688142814244, 7.24913816200202102064102658967, 8.845094165083717309390267109446, 9.552545885528550834848879795168, 10.98191828245050757410511598021, 11.79956167175412017035060281797, 13.18629771784422113562914217308, 14.09932087117300757268139503864
