| L(s) = 1 | + (−0.104 − 0.994i)2-s + (1.04 + 1.15i)3-s + (−0.978 + 0.207i)4-s + (0.550 + 2.16i)5-s + (1.04 − 1.15i)6-s + 3.43·7-s + (0.309 + 0.951i)8-s + (0.0601 − 0.572i)9-s + (2.09 − 0.774i)10-s + (3.68 − 2.67i)11-s + (−1.25 − 0.915i)12-s + (0.214 − 2.04i)13-s + (−0.358 − 3.41i)14-s + (−1.93 + 2.89i)15-s + (0.913 − 0.406i)16-s + (2.37 + 0.505i)17-s + ⋯ |
| L(s) = 1 | + (−0.0739 − 0.703i)2-s + (0.601 + 0.668i)3-s + (−0.489 + 0.103i)4-s + (0.246 + 0.969i)5-s + (0.425 − 0.472i)6-s + 1.29·7-s + (0.109 + 0.336i)8-s + (0.0200 − 0.190i)9-s + (0.663 − 0.244i)10-s + (1.11 − 0.807i)11-s + (−0.363 − 0.264i)12-s + (0.0595 − 0.566i)13-s + (−0.0959 − 0.912i)14-s + (−0.499 + 0.747i)15-s + (0.228 − 0.101i)16-s + (0.576 + 0.122i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0360i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0360i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.25106 - 0.0405777i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.25106 - 0.0405777i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.104 + 0.994i)T \) |
| 5 | \( 1 + (-0.550 - 2.16i)T \) |
| 19 | \( 1 + (4.29 - 0.759i)T \) |
| good | 3 | \( 1 + (-1.04 - 1.15i)T + (-0.313 + 2.98i)T^{2} \) |
| 7 | \( 1 - 3.43T + 7T^{2} \) |
| 11 | \( 1 + (-3.68 + 2.67i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.214 + 2.04i)T + (-12.7 - 2.70i)T^{2} \) |
| 17 | \( 1 + (-2.37 - 0.505i)T + (15.5 + 6.91i)T^{2} \) |
| 23 | \( 1 + (1.17 + 0.522i)T + (15.3 + 17.0i)T^{2} \) |
| 29 | \( 1 + (4.12 - 0.877i)T + (26.4 - 11.7i)T^{2} \) |
| 31 | \( 1 + (-0.827 - 2.54i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-7.14 - 5.18i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-0.317 + 0.141i)T + (27.4 - 30.4i)T^{2} \) |
| 43 | \( 1 + (-2.85 + 4.93i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.29 - 0.275i)T + (42.9 - 19.1i)T^{2} \) |
| 53 | \( 1 + (6.12 - 1.30i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (-3.20 + 1.42i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (1.58 + 0.706i)T + (40.8 + 45.3i)T^{2} \) |
| 67 | \( 1 + (4.78 - 5.30i)T + (-7.00 - 66.6i)T^{2} \) |
| 71 | \( 1 + (-0.868 - 0.964i)T + (-7.42 + 70.6i)T^{2} \) |
| 73 | \( 1 + (-1.65 - 15.7i)T + (-71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (6.71 + 7.45i)T + (-8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (2.06 + 6.35i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-7.40 - 3.29i)T + (59.5 + 66.1i)T^{2} \) |
| 97 | \( 1 + (9.98 + 11.0i)T + (-10.1 + 96.4i)T^{2} \) |
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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
−10.13575741214303020729088514302, −9.307917159873322670135689544591, −8.512855426082522961448226414223, −7.86547371481193213705271364505, −6.56956760743170837928779303667, −5.61489570175329293623337614568, −4.31487988659289168713111487174, −3.62661264213541074628956201285, −2.70120916274110761119579303025, −1.40119825593160582619646163528,
1.36483563491991855074193643816, 2.08116616540426933681461464295, 4.18368580439183769150533989115, 4.69844516361259105568063675962, 5.77579003223748015011262831184, 6.80373310579539276148351233033, 7.78825482400974340060971289664, 8.118779446644874968946253191507, 9.083433578626154292366640372773, 9.551705008935867344069942508702
