| L(s) = 1 | + (0.540 + 0.841i)2-s + (2.16 + 0.310i)3-s + (−0.415 + 0.909i)4-s + (−1.74 + 1.39i)5-s + (0.907 + 1.98i)6-s + (−0.282 + 0.244i)7-s + (−0.989 + 0.142i)8-s + (1.69 + 0.498i)9-s + (−2.11 − 0.714i)10-s + (4.60 + 2.96i)11-s + (−1.18 + 1.83i)12-s + (−3.65 − 3.16i)13-s + (−0.358 − 0.105i)14-s + (−4.20 + 2.47i)15-s + (−0.654 − 0.755i)16-s + (5.87 − 2.68i)17-s + ⋯ |
| L(s) = 1 | + (0.382 + 0.594i)2-s + (1.24 + 0.179i)3-s + (−0.207 + 0.454i)4-s + (−0.781 + 0.624i)5-s + (0.370 + 0.810i)6-s + (−0.106 + 0.0924i)7-s + (−0.349 + 0.0503i)8-s + (0.565 + 0.166i)9-s + (−0.670 − 0.225i)10-s + (1.38 + 0.892i)11-s + (−0.340 + 0.530i)12-s + (−1.01 − 0.877i)13-s + (−0.0957 − 0.0281i)14-s + (−1.08 + 0.639i)15-s + (−0.163 − 0.188i)16-s + (1.42 − 0.650i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.215 - 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.215 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.45021 + 1.16551i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.45021 + 1.16551i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.540 - 0.841i)T \) |
| 5 | \( 1 + (1.74 - 1.39i)T \) |
| 23 | \( 1 + (0.710 + 4.74i)T \) |
| good | 3 | \( 1 + (-2.16 - 0.310i)T + (2.87 + 0.845i)T^{2} \) |
| 7 | \( 1 + (0.282 - 0.244i)T + (0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (-4.60 - 2.96i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (3.65 + 3.16i)T + (1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-5.87 + 2.68i)T + (11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (-0.895 + 1.96i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (2.68 + 5.86i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (-1.32 - 9.23i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (1.00 - 3.40i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (7.81 - 2.29i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (1.08 + 0.155i)T + (41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + 3.02iT - 47T^{2} \) |
| 53 | \( 1 + (-2.40 + 2.08i)T + (7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (-0.366 + 0.422i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (-0.459 - 3.19i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (3.28 + 5.11i)T + (-27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (-9.50 + 6.11i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (2.75 + 1.25i)T + (47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-0.359 + 0.414i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-0.880 + 2.99i)T + (-69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (2.05 - 14.2i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-0.880 - 2.99i)T + (-81.6 + 52.4i)T^{2} \) |
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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
−12.31575790184916949514004018388, −11.88186159517067860311554142833, −10.18031608975446562827159308733, −9.375918620294940194113279290710, −8.245221699967255441345651105513, −7.49548577540119511486815283394, −6.63461442609525732487752079521, −4.88974204437731915107702601659, −3.66852683131047450126845339360, −2.78208399931390440529814665214,
1.59666000192414349408473048414, 3.40019329505936226087876630731, 3.97072215749370181412973901661, 5.57903623777504119396605289037, 7.21899691244781810757597433223, 8.222475556512525476813385712994, 9.103985417980498044742249894863, 9.802299047445140384852610694688, 11.43641744532070272618868354706, 11.97842494847629418127520238098
