| L(s) = 1 | + (−0.107 + 1.41i)2-s + (0.342 − 0.939i)3-s + (−1.97 − 0.302i)4-s + (3.00 − 0.529i)5-s + (1.28 + 0.583i)6-s + (1.26 − 2.18i)7-s + (0.638 − 2.75i)8-s + (−0.766 − 0.642i)9-s + (0.425 + 4.29i)10-s + (−2.96 + 1.71i)11-s + (−0.960 + 1.75i)12-s + (−1.65 − 4.54i)13-s + (2.94 + 2.01i)14-s + (0.529 − 3.00i)15-s + (3.81 + 1.19i)16-s + (2.60 − 2.18i)17-s + ⋯ |
| L(s) = 1 | + (−0.0757 + 0.997i)2-s + (0.197 − 0.542i)3-s + (−0.988 − 0.151i)4-s + (1.34 − 0.237i)5-s + (0.526 + 0.238i)6-s + (0.476 − 0.825i)7-s + (0.225 − 0.974i)8-s + (−0.255 − 0.214i)9-s + (0.134 + 1.35i)10-s + (−0.893 + 0.515i)11-s + (−0.277 + 0.506i)12-s + (−0.458 − 1.26i)13-s + (0.786 + 0.537i)14-s + (0.136 − 0.776i)15-s + (0.954 + 0.298i)16-s + (0.630 − 0.529i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.154i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 + 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.56765 - 0.121943i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.56765 - 0.121943i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.107 - 1.41i)T \) |
| 3 | \( 1 + (-0.342 + 0.939i)T \) |
| 19 | \( 1 + (-0.378 + 4.34i)T \) |
| good | 5 | \( 1 + (-3.00 + 0.529i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-1.26 + 2.18i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.96 - 1.71i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.65 + 4.54i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-2.60 + 2.18i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (0.901 - 5.11i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-6.27 + 7.48i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (4.05 - 7.03i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 11.4iT - 37T^{2} \) |
| 41 | \( 1 + (-7.17 - 2.61i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-3.23 + 0.571i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-7.30 - 6.12i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (0.288 + 0.0508i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (9.52 + 11.3i)T + (-10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (6.55 + 1.15i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (1.55 - 1.85i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.430 - 2.43i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (1.38 + 0.505i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-2.76 - 1.00i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-8.43 - 4.86i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-7.09 + 2.58i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (0.565 - 0.474i)T + (16.8 - 95.5i)T^{2} \) |
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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
−10.67480033630680726631327871365, −9.935033587145819932833386359042, −9.255880038931226335231639993153, −7.898213207318065020728423674431, −7.57652688790769401292976058810, −6.41850924623502722728301620192, −5.41486223104976750007488248637, −4.76528963220755533757247465354, −2.83348493265936577548027241076, −1.08733386259377694886999379002,
1.93744233937916198964539727370, 2.65485167227014363900247324515, 4.13110694645224846254864378413, 5.35260508975782321576751288628, 5.93514891710538607150926521902, 7.77106023240318841982618778849, 8.911069469664625706896739970935, 9.300654226982477642838949902663, 10.47379066212348561892653649172, 10.65649896268719158194982227755
