| L(s) = 1 | + (1.38 + 0.291i)2-s + (−1.80 + 0.908i)3-s + (1.82 + 0.807i)4-s + (1.66 + 3.75i)5-s + (−2.76 + 0.730i)6-s + (0.335 + 0.201i)7-s + (2.29 + 1.65i)8-s + (0.644 − 0.868i)9-s + (1.20 + 5.68i)10-s + (−2.88 − 2.25i)11-s + (−4.03 + 0.204i)12-s + (1.41 − 1.34i)13-s + (0.405 + 0.376i)14-s + (−6.42 − 5.27i)15-s + (2.69 + 2.95i)16-s + (−1.96 − 2.39i)17-s + ⋯ |
| L(s) = 1 | + (0.978 + 0.206i)2-s + (−1.04 + 0.524i)3-s + (0.914 + 0.403i)4-s + (0.745 + 1.68i)5-s + (−1.12 + 0.298i)6-s + (0.126 + 0.0760i)7-s + (0.811 + 0.583i)8-s + (0.214 − 0.289i)9-s + (0.382 + 1.79i)10-s + (−0.870 − 0.679i)11-s + (−1.16 + 0.0589i)12-s + (0.391 − 0.373i)13-s + (0.108 + 0.100i)14-s + (−1.65 − 1.36i)15-s + (0.673 + 0.738i)16-s + (−0.476 − 0.580i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.455 - 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.455 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.05252 + 1.72111i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05252 + 1.72111i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.38 - 0.291i)T \) |
| good | 3 | \( 1 + (1.80 - 0.908i)T + (1.78 - 2.40i)T^{2} \) |
| 5 | \( 1 + (-1.66 - 3.75i)T + (-3.35 + 3.70i)T^{2} \) |
| 7 | \( 1 + (-0.335 - 0.201i)T + (3.29 + 6.17i)T^{2} \) |
| 11 | \( 1 + (2.88 + 2.25i)T + (2.67 + 10.6i)T^{2} \) |
| 13 | \( 1 + (-1.41 + 1.34i)T + (0.637 - 12.9i)T^{2} \) |
| 17 | \( 1 + (1.96 + 2.39i)T + (-3.31 + 16.6i)T^{2} \) |
| 19 | \( 1 + (0.0463 + 0.267i)T + (-17.8 + 6.40i)T^{2} \) |
| 23 | \( 1 + (-0.556 - 0.263i)T + (14.5 + 17.7i)T^{2} \) |
| 29 | \( 1 + (-5.71 - 3.23i)T + (14.9 + 24.8i)T^{2} \) |
| 31 | \( 1 + (0.807 - 0.160i)T + (28.6 - 11.8i)T^{2} \) |
| 37 | \( 1 + (-7.84 - 4.97i)T + (15.8 + 33.4i)T^{2} \) |
| 41 | \( 1 + (5.17 + 5.70i)T + (-4.01 + 40.8i)T^{2} \) |
| 43 | \( 1 + (-2.76 - 8.37i)T + (-34.5 + 25.6i)T^{2} \) |
| 47 | \( 1 + (2.09 + 1.12i)T + (26.1 + 39.0i)T^{2} \) |
| 53 | \( 1 + (-6.98 + 3.95i)T + (27.2 - 45.4i)T^{2} \) |
| 59 | \( 1 + (-6.38 + 6.70i)T + (-2.89 - 58.9i)T^{2} \) |
| 61 | \( 1 + (-0.396 - 5.37i)T + (-60.3 + 8.95i)T^{2} \) |
| 67 | \( 1 + (7.13 + 6.15i)T + (9.83 + 66.2i)T^{2} \) |
| 71 | \( 1 + (-10.0 - 1.49i)T + (67.9 + 20.6i)T^{2} \) |
| 73 | \( 1 + (11.4 - 6.89i)T + (34.4 - 64.3i)T^{2} \) |
| 79 | \( 1 + (2.07 + 0.628i)T + (65.6 + 43.8i)T^{2} \) |
| 83 | \( 1 + (-3.18 + 2.01i)T + (35.4 - 75.0i)T^{2} \) |
| 89 | \( 1 + (-11.8 + 5.59i)T + (56.4 - 68.7i)T^{2} \) |
| 97 | \( 1 + (-2.75 - 1.83i)T + (37.1 + 89.6i)T^{2} \) |
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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
−11.20715809162194601825883393889, −10.57166076057882340617326157805, −10.02047684299611042656807512112, −8.251643420595545474861326721622, −7.08059065600378018950160895688, −6.30206797089108458913144609624, −5.65890739375254577484000576623, −4.82483977500496119778724262250, −3.31426055156597338365847937806, −2.48075351585736543114274864313,
1.03820887550964782766934070590, 2.18300997826262447320434283642, 4.28186678221530996393448386499, 5.00170456510059278842787485847, 5.78460174327524228848125342252, 6.45156494672507422086865948710, 7.70446199478461688845745608687, 8.871957870240240547789753274444, 9.976824414626787751470204720970, 10.83572062198254215325022294440
