| L(s) = 1 | + (0.809 + 0.587i)2-s + (0.913 + 0.406i)3-s + (0.309 + 0.951i)4-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)6-s + (0.346 − 0.384i)7-s + (−0.309 + 0.951i)8-s + (0.669 + 0.743i)9-s + (0.913 − 0.406i)10-s + (1.37 + 0.291i)11-s + (−0.104 + 0.994i)12-s + (0.710 + 6.75i)13-s + (0.506 − 0.107i)14-s + (0.809 − 0.587i)15-s + (−0.809 + 0.587i)16-s + (1.60 − 0.341i)17-s + ⋯ |
| L(s) = 1 | + (0.572 + 0.415i)2-s + (0.527 + 0.234i)3-s + (0.154 + 0.475i)4-s + (0.223 − 0.387i)5-s + (0.204 + 0.353i)6-s + (0.130 − 0.145i)7-s + (−0.109 + 0.336i)8-s + (0.223 + 0.247i)9-s + (0.288 − 0.128i)10-s + (0.413 + 0.0879i)11-s + (−0.0301 + 0.287i)12-s + (0.196 + 1.87i)13-s + (0.135 − 0.0287i)14-s + (0.208 − 0.151i)15-s + (−0.202 + 0.146i)16-s + (0.389 − 0.0828i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.494 - 0.868i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.494 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.43505 + 1.41558i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.43505 + 1.41558i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.809 - 0.587i)T \) |
| 3 | \( 1 + (-0.913 - 0.406i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 31 | \( 1 + (-5.20 - 1.96i)T \) |
| good | 7 | \( 1 + (-0.346 + 0.384i)T + (-0.731 - 6.96i)T^{2} \) |
| 11 | \( 1 + (-1.37 - 0.291i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (-0.710 - 6.75i)T + (-12.7 + 2.70i)T^{2} \) |
| 17 | \( 1 + (-1.60 + 0.341i)T + (15.5 - 6.91i)T^{2} \) |
| 19 | \( 1 + (-0.750 + 7.13i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (1.43 - 4.41i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (2.09 + 1.52i)T + (8.96 + 27.5i)T^{2} \) |
| 37 | \( 1 + (-4.52 - 7.84i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-6.59 + 2.93i)T + (27.4 - 30.4i)T^{2} \) |
| 43 | \( 1 + (-0.865 + 8.23i)T + (-42.0 - 8.94i)T^{2} \) |
| 47 | \( 1 + (-3.10 + 2.25i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (9.00 + 9.99i)T + (-5.54 + 52.7i)T^{2} \) |
| 59 | \( 1 + (-1.91 - 0.850i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 + (3.25 - 5.63i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (2.08 + 2.31i)T + (-7.42 + 70.6i)T^{2} \) |
| 73 | \( 1 + (9.52 + 2.02i)T + (66.6 + 29.6i)T^{2} \) |
| 79 | \( 1 + (11.1 - 2.36i)T + (72.1 - 32.1i)T^{2} \) |
| 83 | \( 1 + (8.74 - 3.89i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (4.38 + 13.4i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (2.17 + 6.69i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−9.970646356922134927902497374700, −9.186091773731089112903983848611, −8.686109377910499820410798770534, −7.51782866339875510998603828572, −6.84105674953171208890910139191, −5.88050786202353265175990851527, −4.66515302709849768269683420620, −4.21108026369462383364419821394, −2.94492284946160223315116385083, −1.65688134857502433185578519818,
1.22157177829214919914774361230, 2.59903079324904815515436323837, 3.36085331752743379279274028550, 4.39075539714513640204114784498, 5.80300574649406854242163764327, 6.11392156139605782418307375057, 7.59740937096007670567067269802, 8.062805622398921410790473132107, 9.220811538086665195309417152842, 10.14045086453118011613633367510
