| L(s) = 1 | + (0.104 − 0.994i)2-s + (0.669 + 0.743i)3-s + (−0.978 − 0.207i)4-s + (3.80 + 1.69i)5-s + (0.809 − 0.587i)6-s + (−2.12 + 1.57i)7-s + (−0.309 + 0.951i)8-s + (−0.104 + 0.994i)9-s + (2.08 − 3.60i)10-s + (1.31 + 3.04i)11-s + (−0.5 − 0.866i)12-s + (−1.76 − 1.28i)13-s + (1.34 + 2.27i)14-s + (1.28 + 3.95i)15-s + (0.913 + 0.406i)16-s + (0.0909 + 0.865i)17-s + ⋯ |
| L(s) = 1 | + (0.0739 − 0.703i)2-s + (0.386 + 0.429i)3-s + (−0.489 − 0.103i)4-s + (1.70 + 0.757i)5-s + (0.330 − 0.239i)6-s + (−0.802 + 0.597i)7-s + (−0.109 + 0.336i)8-s + (−0.0348 + 0.331i)9-s + (0.658 − 1.14i)10-s + (0.396 + 0.918i)11-s + (−0.144 − 0.249i)12-s + (−0.489 − 0.355i)13-s + (0.360 + 0.608i)14-s + (0.332 + 1.02i)15-s + (0.228 + 0.101i)16-s + (0.0220 + 0.209i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 - 0.370i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.928 - 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.82553 + 0.350619i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.82553 + 0.350619i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.104 + 0.994i)T \) |
| 3 | \( 1 + (-0.669 - 0.743i)T \) |
| 7 | \( 1 + (2.12 - 1.57i)T \) |
| 11 | \( 1 + (-1.31 - 3.04i)T \) |
| good | 5 | \( 1 + (-3.80 - 1.69i)T + (3.34 + 3.71i)T^{2} \) |
| 13 | \( 1 + (1.76 + 1.28i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.0909 - 0.865i)T + (-16.6 + 3.53i)T^{2} \) |
| 19 | \( 1 + (1.13 - 0.241i)T + (17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (1.42 + 2.47i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.53 + 4.71i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (1.62 - 0.725i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (-6.57 + 7.30i)T + (-3.86 - 36.7i)T^{2} \) |
| 41 | \( 1 + (-2.10 + 6.48i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 7.87T + 43T^{2} \) |
| 47 | \( 1 + (-7.39 + 1.57i)T + (42.9 - 19.1i)T^{2} \) |
| 53 | \( 1 + (-5.01 + 2.23i)T + (35.4 - 39.3i)T^{2} \) |
| 59 | \( 1 + (7.96 + 1.69i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (-0.669 - 0.298i)T + (40.8 + 45.3i)T^{2} \) |
| 67 | \( 1 + (3.89 - 6.73i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (4.47 - 3.25i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (1.68 + 0.359i)T + (66.6 + 29.6i)T^{2} \) |
| 79 | \( 1 + (-1.66 + 15.8i)T + (-77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (0.738 - 0.536i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (6.98 + 12.1i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-12.0 - 8.77i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.68986033508769722326500892790, −10.20740961684421585884305388756, −9.428302172187154974281625989961, −9.048043753468348426091404090557, −7.37222868429682906633511680350, −6.19371124055928664863175001278, −5.49213206790036554630437379227, −4.06948015155710353069192644950, −2.66890232088601564530628327987, −2.13959598885050272984649682925,
1.19838876753380805180569706864, 2.82306219852533879860650424361, 4.34651061041858546205633829872, 5.65721532971062575773695902604, 6.25527376748907759418904063380, 7.14173372624672482036658703764, 8.347815384254097338062090265788, 9.376888706011305654899203816686, 9.546474948862635916905942480411, 10.77989641954689117479827886010
