| L(s) = 1 | + (0.173 + 0.984i)2-s + (−0.0317 + 0.180i)3-s + (−0.939 + 0.342i)4-s + (0.766 + 0.642i)5-s − 0.182·6-s + (1.13 + 0.952i)7-s + (−0.5 − 0.866i)8-s + (2.78 + 1.01i)9-s + (−0.5 + 0.866i)10-s + (−0.0395 − 0.0685i)11-s + (−0.0317 − 0.180i)12-s + (−1.35 + 0.492i)13-s + (−0.740 + 1.28i)14-s + (−0.140 + 0.117i)15-s + (0.766 − 0.642i)16-s + (−0.0477 − 0.0173i)17-s + ⋯ |
| L(s) = 1 | + (0.122 + 0.696i)2-s + (−0.0183 + 0.103i)3-s + (−0.469 + 0.171i)4-s + (0.342 + 0.287i)5-s − 0.0746·6-s + (0.428 + 0.359i)7-s + (−0.176 − 0.306i)8-s + (0.929 + 0.338i)9-s + (−0.158 + 0.273i)10-s + (−0.0119 − 0.0206i)11-s + (−0.00916 − 0.0519i)12-s + (−0.375 + 0.136i)13-s + (−0.197 + 0.342i)14-s + (−0.0361 + 0.0303i)15-s + (0.191 − 0.160i)16-s + (−0.0115 − 0.00421i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.116 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.116 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.00347 + 1.12751i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.00347 + 1.12751i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.173 - 0.984i)T \) |
| 5 | \( 1 + (-0.766 - 0.642i)T \) |
| 37 | \( 1 + (-0.570 + 6.05i)T \) |
| good | 3 | \( 1 + (0.0317 - 0.180i)T + (-2.81 - 1.02i)T^{2} \) |
| 7 | \( 1 + (-1.13 - 0.952i)T + (1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (0.0395 + 0.0685i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.35 - 0.492i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (0.0477 + 0.0173i)T + (13.0 + 10.9i)T^{2} \) |
| 19 | \( 1 + (0.860 - 4.88i)T + (-17.8 - 6.49i)T^{2} \) |
| 23 | \( 1 + (1.14 - 1.98i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.47 - 2.56i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 0.605T + 31T^{2} \) |
| 41 | \( 1 + (-6.43 + 2.34i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + 8.69T + 43T^{2} \) |
| 47 | \( 1 + (-4.53 + 7.85i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.793 + 0.666i)T + (9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (-0.696 + 0.584i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-9.99 + 3.63i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (1.93 + 1.62i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.29 + 7.31i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + 9.62T + 73T^{2} \) |
| 79 | \( 1 + (5.04 + 4.23i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-0.685 - 0.249i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (-10.8 + 9.12i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-1.01 + 1.76i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.74531588741692889333949423121, −10.50809101865226951607667167036, −9.831854575779251434486032963304, −8.767230320263983007687757460229, −7.76284384074064649474924842016, −6.95476654406028268859606967103, −5.83342810197030291726462860439, −4.91495109703447278469413663178, −3.74992986288667009649288965967, −1.95561494730454181526005157334,
1.13153058792138832002442735000, 2.58768679108223922831209188929, 4.17949693229827838267375571839, 4.91621132159683937842792064141, 6.31451563799491957994857960335, 7.40851539654391493911588008009, 8.523808777641411897651123881498, 9.564004969291127268668275112658, 10.22012373113561550703668906099, 11.17548876339101993713858922629
