| L(s) = 1 | + (−0.621 − 0.263i)2-s + (1.68 − 0.415i)3-s + (−1.07 − 1.10i)4-s + (1.25 + 0.885i)5-s + (−1.15 − 0.184i)6-s + (−4.15 − 1.60i)7-s + (0.864 + 2.23i)8-s + (2.65 − 1.39i)9-s + (−0.544 − 0.879i)10-s + (0.491 + 0.650i)11-s + (−2.26 − 1.41i)12-s + (−0.978 − 6.30i)13-s + (2.15 + 2.09i)14-s + (2.47 + 0.969i)15-s + (−0.0454 + 1.47i)16-s + (0.691 − 0.243i)17-s + ⋯ |
| L(s) = 1 | + (−0.439 − 0.185i)2-s + (0.970 − 0.239i)3-s + (−0.537 − 0.554i)4-s + (0.559 + 0.395i)5-s + (−0.471 − 0.0752i)6-s + (−1.56 − 0.607i)7-s + (0.305 + 0.788i)8-s + (0.885 − 0.465i)9-s + (−0.172 − 0.278i)10-s + (0.148 + 0.196i)11-s + (−0.654 − 0.409i)12-s + (−0.271 − 1.74i)13-s + (0.576 + 0.558i)14-s + (0.638 + 0.250i)15-s + (−0.0113 + 0.368i)16-s + (0.167 − 0.0591i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.854 + 0.520i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.854 + 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.257028 - 0.916360i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.257028 - 0.916360i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.68 + 0.415i)T \) |
| 103 | \( 1 + (9.99 - 1.78i)T \) |
| good | 2 | \( 1 + (0.621 + 0.263i)T + (1.39 + 1.43i)T^{2} \) |
| 5 | \( 1 + (-1.25 - 0.885i)T + (1.66 + 4.71i)T^{2} \) |
| 7 | \( 1 + (4.15 + 1.60i)T + (5.17 + 4.71i)T^{2} \) |
| 11 | \( 1 + (-0.491 - 0.650i)T + (-3.01 + 10.5i)T^{2} \) |
| 13 | \( 1 + (0.978 + 6.30i)T + (-12.3 + 3.94i)T^{2} \) |
| 17 | \( 1 + (-0.691 + 0.243i)T + (13.2 - 10.6i)T^{2} \) |
| 19 | \( 1 + (1.61 + 0.0994i)T + (18.8 + 2.33i)T^{2} \) |
| 23 | \( 1 + (0.822 - 6.64i)T + (-22.3 - 5.61i)T^{2} \) |
| 29 | \( 1 + (4.12 + 8.96i)T + (-18.8 + 22.0i)T^{2} \) |
| 31 | \( 1 + (8.71 + 0.268i)T + (30.9 + 1.90i)T^{2} \) |
| 37 | \( 1 + (-1.26 + 1.39i)T + (-3.41 - 36.8i)T^{2} \) |
| 41 | \( 1 + (-4.27 + 0.396i)T + (40.3 - 7.53i)T^{2} \) |
| 43 | \( 1 + (-2.16 + 6.81i)T + (-35.0 - 24.8i)T^{2} \) |
| 47 | \( 1 + 11.8T + 47T^{2} \) |
| 53 | \( 1 + (4.96 - 7.49i)T + (-20.6 - 48.8i)T^{2} \) |
| 59 | \( 1 + (0.940 + 2.42i)T + (-43.6 + 39.7i)T^{2} \) |
| 61 | \( 1 + (4.01 - 4.68i)T + (-9.35 - 60.2i)T^{2} \) |
| 67 | \( 1 + (-4.94 - 0.768i)T + (63.8 + 20.3i)T^{2} \) |
| 71 | \( 1 + (-6.39 + 4.53i)T + (23.5 - 66.9i)T^{2} \) |
| 73 | \( 1 + (-5.98 + 0.554i)T + (71.7 - 13.4i)T^{2} \) |
| 79 | \( 1 + (-1.56 - 1.10i)T + (26.2 + 74.5i)T^{2} \) |
| 83 | \( 1 + (-13.5 + 2.09i)T + (79.0 - 25.1i)T^{2} \) |
| 89 | \( 1 + (1.60 + 5.62i)T + (-75.6 + 46.8i)T^{2} \) |
| 97 | \( 1 + (8.58 + 10.0i)T + (-14.8 + 95.8i)T^{2} \) |
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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
−9.619281929558845510786404435677, −9.328320241618239218923272235195, −8.021146507132976547469329346217, −7.42805917417953475022390377428, −6.28718465392389897655286116199, −5.56620346622091788879068746518, −4.00784046474839767674715312603, −3.16362917300304087122867938269, −2.03787914775547856968013226578, −0.44570009966146904874578131556,
1.90584840484907786641178289038, 3.17032387627826809495415197417, 3.94888389512891916380423104884, 5.01716162259911449604348309070, 6.46075414887244848811492825608, 7.02456067410604425007645845887, 8.210609336626016871036254830726, 9.044540206347286751818750652998, 9.374899494009531435064774978844, 9.753862465990099987645306871670
