| 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 + (0.978 + 1.08i)7-s + (0.309 + 0.951i)8-s + (0.669 − 0.743i)9-s + (−0.913 − 0.406i)10-s + (1.01 − 0.215i)11-s + (−0.104 − 0.994i)12-s + (−0.241 + 2.30i)13-s + (−1.43 − 0.304i)14-s + (0.809 + 0.587i)15-s + (−0.809 − 0.587i)16-s + (1.15 + 0.244i)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 + (0.369 + 0.410i)7-s + (0.109 + 0.336i)8-s + (0.223 − 0.247i)9-s + (−0.288 − 0.128i)10-s + (0.305 − 0.0649i)11-s + (−0.0301 − 0.287i)12-s + (−0.0670 + 0.638i)13-s + (−0.382 − 0.0812i)14-s + (0.208 + 0.151i)15-s + (−0.202 − 0.146i)16-s + (0.279 + 0.0593i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.486 - 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.486 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.34269 + 0.789431i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.34269 + 0.789431i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-4.38 - 3.43i)T \) |
| good | 7 | \( 1 + (-0.978 - 1.08i)T + (-0.731 + 6.96i)T^{2} \) |
| 11 | \( 1 + (-1.01 + 0.215i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (0.241 - 2.30i)T + (-12.7 - 2.70i)T^{2} \) |
| 17 | \( 1 + (-1.15 - 0.244i)T + (15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (-0.720 - 6.85i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (1.62 + 4.99i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-2.58 + 1.87i)T + (8.96 - 27.5i)T^{2} \) |
| 37 | \( 1 + (3.71 - 6.44i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.43 - 1.52i)T + (27.4 + 30.4i)T^{2} \) |
| 43 | \( 1 + (0.199 + 1.89i)T + (-42.0 + 8.94i)T^{2} \) |
| 47 | \( 1 + (0.374 + 0.272i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (2.94 - 3.27i)T + (-5.54 - 52.7i)T^{2} \) |
| 59 | \( 1 + (6.65 - 2.96i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 - 7.09T + 61T^{2} \) |
| 67 | \( 1 + (-3.51 - 6.08i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-10.8 + 12.0i)T + (-7.42 - 70.6i)T^{2} \) |
| 73 | \( 1 + (-10.0 + 2.13i)T + (66.6 - 29.6i)T^{2} \) |
| 79 | \( 1 + (-3.80 - 0.808i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (-5.72 - 2.54i)T + (55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 + (2.27 - 6.99i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (4.37 - 13.4i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.03505246293856446258828290614, −9.320056861029000663074882829456, −8.340579615836474039935547172834, −7.966052396350646008749456202468, −6.76196372618526902890124591003, −6.24247424486519919630570983054, −5.08958886749592497462868746087, −3.83834347890149270441873438763, −2.51993965705501633649458520122, −1.45381808967758181751740830936,
0.933556858038732504443952175782, 2.27991865555870511251429737008, 3.38888278446412877453336015601, 4.44947787910477034198276542765, 5.40966008181309050400062098983, 6.77019314278261362598721409648, 7.67372582784894571812937291716, 8.323966512449191898511996022317, 9.289553261325654124190319650334, 9.701289901059327708876661043486
