| L(s) = 1 | + (0.0713 − 0.997i)2-s + (2.04 − 0.445i)3-s + (−0.989 − 0.142i)4-s + (1.87 − 1.21i)5-s + (−0.298 − 2.07i)6-s + (−0.714 + 0.390i)7-s + (−0.212 + 0.977i)8-s + (1.26 − 0.579i)9-s + (−1.07 − 1.96i)10-s + (0.227 + 0.197i)11-s + (−2.09 + 0.149i)12-s + (−1.18 + 2.16i)13-s + (0.338 + 0.740i)14-s + (3.30 − 3.32i)15-s + (0.959 + 0.281i)16-s + (−3.11 − 2.33i)17-s + ⋯ |
| L(s) = 1 | + (0.0504 − 0.705i)2-s + (1.18 − 0.257i)3-s + (−0.494 − 0.0711i)4-s + (0.839 − 0.542i)5-s + (−0.121 − 0.847i)6-s + (−0.270 + 0.147i)7-s + (−0.0751 + 0.345i)8-s + (0.422 − 0.193i)9-s + (−0.340 − 0.619i)10-s + (0.0685 + 0.0594i)11-s + (−0.603 + 0.0431i)12-s + (−0.328 + 0.600i)13-s + (0.0903 + 0.197i)14-s + (0.853 − 0.857i)15-s + (0.239 + 0.0704i)16-s + (−0.756 − 0.566i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.327 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.327 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.45190 - 1.03296i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.45190 - 1.03296i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.0713 + 0.997i)T \) |
| 5 | \( 1 + (-1.87 + 1.21i)T \) |
| 23 | \( 1 + (-4.11 + 2.45i)T \) |
| good | 3 | \( 1 + (-2.04 + 0.445i)T + (2.72 - 1.24i)T^{2} \) |
| 7 | \( 1 + (0.714 - 0.390i)T + (3.78 - 5.88i)T^{2} \) |
| 11 | \( 1 + (-0.227 - 0.197i)T + (1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (1.18 - 2.16i)T + (-7.02 - 10.9i)T^{2} \) |
| 17 | \( 1 + (3.11 + 2.33i)T + (4.78 + 16.3i)T^{2} \) |
| 19 | \( 1 + (0.734 - 5.11i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (2.53 - 0.364i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (1.50 - 0.967i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (0.689 + 0.257i)T + (27.9 + 24.2i)T^{2} \) |
| 41 | \( 1 + (0.0867 - 0.189i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (2.18 + 10.0i)T + (-39.1 + 17.8i)T^{2} \) |
| 47 | \( 1 + (-8.33 - 8.33i)T + 47iT^{2} \) |
| 53 | \( 1 + (-4.39 - 8.05i)T + (-28.6 + 44.5i)T^{2} \) |
| 59 | \( 1 + (-0.263 - 0.896i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (2.69 + 4.19i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-9.87 - 0.705i)T + (66.3 + 9.53i)T^{2} \) |
| 71 | \( 1 + (-5.22 - 6.02i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (2.91 + 3.89i)T + (-20.5 + 70.0i)T^{2} \) |
| 79 | \( 1 + (11.3 - 3.32i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (-0.392 + 1.05i)T + (-62.7 - 54.3i)T^{2} \) |
| 89 | \( 1 + (11.4 + 7.38i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (4.27 + 11.4i)T + (-73.3 + 63.5i)T^{2} \) |
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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
−12.33206147661337520590372598803, −11.00816751493169250083484825021, −9.835758543149761580648465003797, −9.105580440653991450973348470057, −8.511892177835838928461859697524, −7.13248522473149864756675227799, −5.67295293885227857684374438318, −4.30241811629319664495840867252, −2.83189752629082507265660671861, −1.81459740442649134052427742107,
2.48030604816635136006562899559, 3.63760218513569806405643957403, 5.19195140525480506140094221778, 6.45586968132259489093796684361, 7.37952817489731644770800263499, 8.577170504540811571301153054164, 9.293939149022477806460070000247, 10.11492656639476786508946792945, 11.24727891921979698098559498829, 13.10896300114747475664639250425
