| L(s) = 1 | + (0.750 + 1.47i)2-s + (−0.248 + 1.71i)3-s + (−0.432 + 0.594i)4-s + (0.103 + 0.651i)5-s + (−2.71 + 0.920i)6-s + (−1.17 − 1.38i)7-s + (2.06 + 0.327i)8-s + (−2.87 − 0.853i)9-s + (−0.882 + 0.641i)10-s + (0.889 − 1.45i)11-s + (−0.911 − 0.888i)12-s + (−0.0823 − 1.04i)13-s + (1.14 − 2.77i)14-s + (−1.14 + 0.0147i)15-s + (1.52 + 4.68i)16-s + (0.540 + 2.25i)17-s + ⋯ |
| L(s) = 1 | + (0.530 + 1.04i)2-s + (−0.143 + 0.989i)3-s + (−0.216 + 0.297i)4-s + (0.0461 + 0.291i)5-s + (−1.10 + 0.375i)6-s + (−0.445 − 0.521i)7-s + (0.730 + 0.115i)8-s + (−0.958 − 0.284i)9-s + (−0.279 + 0.202i)10-s + (0.268 − 0.437i)11-s + (−0.263 − 0.256i)12-s + (−0.0228 − 0.290i)13-s + (0.307 − 0.741i)14-s + (−0.295 + 0.00381i)15-s + (0.380 + 1.17i)16-s + (0.131 + 0.546i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.351 - 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.351 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.760898 + 1.09879i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.760898 + 1.09879i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.248 - 1.71i)T \) |
| 41 | \( 1 + (6.39 + 0.327i)T \) |
| good | 2 | \( 1 + (-0.750 - 1.47i)T + (-1.17 + 1.61i)T^{2} \) |
| 5 | \( 1 + (-0.103 - 0.651i)T + (-4.75 + 1.54i)T^{2} \) |
| 7 | \( 1 + (1.17 + 1.38i)T + (-1.09 + 6.91i)T^{2} \) |
| 11 | \( 1 + (-0.889 + 1.45i)T + (-4.99 - 9.80i)T^{2} \) |
| 13 | \( 1 + (0.0823 + 1.04i)T + (-12.8 + 2.03i)T^{2} \) |
| 17 | \( 1 + (-0.540 - 2.25i)T + (-15.1 + 7.71i)T^{2} \) |
| 19 | \( 1 + (-0.405 + 5.14i)T + (-18.7 - 2.97i)T^{2} \) |
| 23 | \( 1 + (1.68 - 5.19i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-0.192 + 0.799i)T + (-25.8 - 13.1i)T^{2} \) |
| 31 | \( 1 + (3.67 + 5.05i)T + (-9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-4.58 - 3.33i)T + (11.4 + 35.1i)T^{2} \) |
| 43 | \( 1 + (7.88 - 4.01i)T + (25.2 - 34.7i)T^{2} \) |
| 47 | \( 1 + (5.81 + 4.96i)T + (7.35 + 46.4i)T^{2} \) |
| 53 | \( 1 + (-2.73 - 0.655i)T + (47.2 + 24.0i)T^{2} \) |
| 59 | \( 1 + (12.0 + 3.91i)T + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (5.93 - 11.6i)T + (-35.8 - 49.3i)T^{2} \) |
| 67 | \( 1 + (6.07 + 9.91i)T + (-30.4 + 59.6i)T^{2} \) |
| 71 | \( 1 + (-8.13 - 4.98i)T + (32.2 + 63.2i)T^{2} \) |
| 73 | \( 1 + (-3.53 - 3.53i)T + 73iT^{2} \) |
| 79 | \( 1 + (-2.58 - 6.24i)T + (-55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 - 6.34iT - 83T^{2} \) |
| 89 | \( 1 + (7.77 - 6.64i)T + (13.9 - 87.9i)T^{2} \) |
| 97 | \( 1 + (-8.27 + 5.07i)T + (44.0 - 86.4i)T^{2} \) |
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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
−13.94680486282388806678350507484, −13.20561248559488135052246648768, −11.48097721284863997716248659096, −10.59997653223332455350407751599, −9.625409533872373617320661952080, −8.238170175250512855950481262127, −6.88321875547449640218063550403, −5.91404928893665429824584583155, −4.76282195032873303752196793529, −3.46811991752839670982221047556,
1.75789966507582353527062332059, 3.17139240487403983442431435009, 4.91948366160606021747356681918, 6.39877957915854956854482948060, 7.60448006182388000313258998954, 8.936192955756550781063092081488, 10.28799580047943322193894028542, 11.45598231724759327757693834827, 12.40790896486091691002411686861, 12.60385867854966885738349683127
