| L(s) = 1 | + (−0.540 − 0.841i)2-s + (−2.16 − 0.310i)3-s + (−0.415 + 0.909i)4-s + (−2.22 − 0.230i)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 + (1.00 + 1.99i)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 + (4.73 + 1.18i)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 + (−0.994 − 0.103i)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 + (0.318 + 0.631i)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 + (1.22 + 0.307i)15-s + (−0.163 − 0.188i)16-s + (−1.42 + 0.650i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.838 - 0.544i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.838 - 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.456866 + 0.135377i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.456866 + 0.135377i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.22 + 0.230i)T \) |
| 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
−11.93916578851440677906095584752, −11.36019753695872503797313701778, −10.87904903989172769128531403115, −9.386345243972222223026882825248, −8.557805464452228465990225898290, −7.09959208974791746514281626419, −6.40442695172633568499423375414, −4.67901577310971850216287143717, −3.83807725105967881586489998544, −1.41307451136279355654437961346,
0.58980086463494942367848424443, 3.72650750553923834679705547174, 4.96745650825546917094704035350, 6.16482461133755836187579912410, 6.79932050275054644842270320200, 8.236780193738516469041431233788, 8.942381182987730565344981861716, 10.45010429583293484833520058491, 11.32381623516272199828266296837, 11.62090975568845108568650195133
