| L(s) = 1 | + (0.654 − 0.755i)2-s + (−0.512 + 0.329i)3-s + (−0.142 − 0.989i)4-s + (0.154 + 0.339i)5-s + (−0.0867 + 0.603i)6-s + (−1.97 + 0.580i)7-s + (−0.841 − 0.540i)8-s + (−1.09 + 2.39i)9-s + (0.357 + 0.105i)10-s + (−1.61 − 1.86i)11-s + (0.398 + 0.460i)12-s + (3.58 + 1.05i)13-s + (−0.855 + 1.87i)14-s + (−0.191 − 0.122i)15-s + (−0.959 + 0.281i)16-s + (0.897 − 6.24i)17-s + ⋯ |
| L(s) = 1 | + (0.463 − 0.534i)2-s + (−0.295 + 0.190i)3-s + (−0.0711 − 0.494i)4-s + (0.0692 + 0.151i)5-s + (−0.0353 + 0.246i)6-s + (−0.746 + 0.219i)7-s + (−0.297 − 0.191i)8-s + (−0.364 + 0.797i)9-s + (0.113 + 0.0332i)10-s + (−0.486 − 0.561i)11-s + (0.115 + 0.132i)12-s + (0.993 + 0.291i)13-s + (−0.228 + 0.500i)14-s + (−0.0493 − 0.0317i)15-s + (−0.239 + 0.0704i)16-s + (0.217 − 1.51i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.846533 - 0.212531i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.846533 - 0.212531i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.654 + 0.755i)T \) |
| 23 | \( 1 + (1.76 + 4.45i)T \) |
| good | 3 | \( 1 + (0.512 - 0.329i)T + (1.24 - 2.72i)T^{2} \) |
| 5 | \( 1 + (-0.154 - 0.339i)T + (-3.27 + 3.77i)T^{2} \) |
| 7 | \( 1 + (1.97 - 0.580i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (1.61 + 1.86i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-3.58 - 1.05i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.897 + 6.24i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.468 - 3.25i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (1.01 - 7.06i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (-4.91 - 3.16i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (1.94 - 4.25i)T + (-24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (1.98 + 4.34i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (9.36 - 6.02i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 - 2.44T + 47T^{2} \) |
| 53 | \( 1 + (5.80 - 1.70i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (-11.3 - 3.32i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-1.41 - 0.911i)T + (25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-9.23 + 10.6i)T + (-9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (2.64 - 3.05i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (1.23 + 8.60i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (-2.28 - 0.671i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (-1.76 + 3.85i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (7.59 - 4.88i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (4.69 + 10.2i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−16.00467903080045150476659415575, −14.25347537169464405244880275644, −13.48215621859116067649079190321, −12.19118861382314977979457340751, −11.03177934837162357369287280843, −10.07008996297020136706498821387, −8.498206266573244034292850666950, −6.43822082882635961482822196777, −5.06548363283381029916753139471, −3.03495215890506590148935069652,
3.66207636302238012715618758039, 5.68087471540256896321167036375, 6.75725296116448395531755582806, 8.345211885809754065988655976616, 9.839815841441828476076045797500, 11.43674514253753380056394870858, 12.76412615800866275214745368584, 13.40842828719453163265253412686, 14.99277120553979096595313540282, 15.72295873068025060036088082328
