| L(s) = 1 | + (−0.422 − 0.906i)2-s + (1.61 + 0.614i)3-s + (−0.642 + 0.766i)4-s + (1.05 + 1.97i)5-s + (−0.126 − 1.72i)6-s + (−0.0450 − 0.168i)7-s + (0.965 + 0.258i)8-s + (2.24 + 1.99i)9-s + (1.34 − 1.78i)10-s + (1.02 − 0.592i)11-s + (−1.51 + 0.845i)12-s + (−0.862 + 0.604i)13-s + (−0.133 + 0.111i)14-s + (0.494 + 3.84i)15-s + (−0.173 − 0.984i)16-s + (−0.00890 − 0.0191i)17-s + ⋯ |
| L(s) = 1 | + (−0.298 − 0.640i)2-s + (0.934 + 0.355i)3-s + (−0.321 + 0.383i)4-s + (0.471 + 0.881i)5-s + (−0.0518 − 0.705i)6-s + (−0.0170 − 0.0636i)7-s + (0.341 + 0.0915i)8-s + (0.747 + 0.663i)9-s + (0.424 − 0.565i)10-s + (0.309 − 0.178i)11-s + (−0.436 + 0.243i)12-s + (−0.239 + 0.167i)13-s + (−0.0356 + 0.0299i)14-s + (0.127 + 0.991i)15-s + (−0.0434 − 0.246i)16-s + (−0.00216 − 0.00463i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.924 - 0.380i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.924 - 0.380i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.76925 + 0.349653i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.76925 + 0.349653i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.422 + 0.906i)T \) |
| 3 | \( 1 + (-1.61 - 0.614i)T \) |
| 5 | \( 1 + (-1.05 - 1.97i)T \) |
| 19 | \( 1 + (3.61 - 2.43i)T \) |
| good | 7 | \( 1 + (0.0450 + 0.168i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (-1.02 + 0.592i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.862 - 0.604i)T + (4.44 - 12.2i)T^{2} \) |
| 17 | \( 1 + (0.00890 + 0.0191i)T + (-10.9 + 13.0i)T^{2} \) |
| 23 | \( 1 + (-5.33 + 0.467i)T + (22.6 - 3.99i)T^{2} \) |
| 29 | \( 1 + (-6.37 + 2.31i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (2.85 - 4.94i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.29 + 2.29i)T - 37iT^{2} \) |
| 41 | \( 1 + (-1.39 + 0.245i)T + (38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (8.71 + 0.762i)T + (42.3 + 7.46i)T^{2} \) |
| 47 | \( 1 + (3.72 + 1.73i)T + (30.2 + 36.0i)T^{2} \) |
| 53 | \( 1 + (0.422 - 0.0369i)T + (52.1 - 9.20i)T^{2} \) |
| 59 | \( 1 + (-6.34 - 2.30i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (3.04 + 2.55i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-6.76 + 14.5i)T + (-43.0 - 51.3i)T^{2} \) |
| 71 | \( 1 + (3.95 + 4.71i)T + (-12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (-3.74 + 5.35i)T + (-24.9 - 68.5i)T^{2} \) |
| 79 | \( 1 + (10.5 - 1.85i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (1.55 + 5.79i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (-0.687 + 3.89i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-10.7 + 4.99i)T + (62.3 - 74.3i)T^{2} \) |
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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
−10.53103606938158296591656816002, −10.04456126300674303260502595495, −9.132545641012762002263171134348, −8.415165237583978292909534182595, −7.35396075670409078758070071310, −6.46059215465067210371076974593, −4.92974360520013076248305078230, −3.73618009349949664028848723832, −2.87000629566201081927149523899, −1.81642519937153755379500727450,
1.16139877533388607096873634237, 2.55567069377754204208067343537, 4.16671961232186659094261577266, 5.12726204453784645740805541804, 6.36294330569974585508648753470, 7.15109445759109256355375087747, 8.229015111849309994185239334891, 8.790566490261892087835605381240, 9.499183325970735046822379961793, 10.26704787189303791693229268807
