| L(s) = 1 | + (0.175 − 1.40i)2-s + (1.73 + 1.42i)3-s + (−1.93 − 0.492i)4-s + (−0.314 − 1.03i)5-s + (2.29 − 2.18i)6-s + (0.980 + 0.195i)7-s + (−1.03 + 2.63i)8-s + (0.393 + 1.98i)9-s + (−1.51 + 0.259i)10-s + (−0.504 + 5.11i)11-s + (−2.65 − 3.60i)12-s + (2.72 + 0.825i)13-s + (0.445 − 1.34i)14-s + (0.929 − 2.24i)15-s + (3.51 + 1.91i)16-s + (1.23 + 2.98i)17-s + ⋯ |
| L(s) = 1 | + (0.124 − 0.992i)2-s + (0.999 + 0.820i)3-s + (−0.969 − 0.246i)4-s + (−0.140 − 0.463i)5-s + (0.938 − 0.890i)6-s + (0.370 + 0.0737i)7-s + (−0.364 + 0.931i)8-s + (0.131 + 0.660i)9-s + (−0.477 + 0.0820i)10-s + (−0.151 + 1.54i)11-s + (−0.766 − 1.04i)12-s + (0.754 + 0.228i)13-s + (0.119 − 0.358i)14-s + (0.239 − 0.579i)15-s + (0.878 + 0.477i)16-s + (0.299 + 0.723i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.231i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 + 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.17140 - 0.254451i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.17140 - 0.254451i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.175 + 1.40i)T \) |
| 7 | \( 1 + (-0.980 - 0.195i)T \) |
| good | 3 | \( 1 + (-1.73 - 1.42i)T + (0.585 + 2.94i)T^{2} \) |
| 5 | \( 1 + (0.314 + 1.03i)T + (-4.15 + 2.77i)T^{2} \) |
| 11 | \( 1 + (0.504 - 5.11i)T + (-10.7 - 2.14i)T^{2} \) |
| 13 | \( 1 + (-2.72 - 0.825i)T + (10.8 + 7.22i)T^{2} \) |
| 17 | \( 1 + (-1.23 - 2.98i)T + (-12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (-0.960 - 1.79i)T + (-10.5 + 15.7i)T^{2} \) |
| 23 | \( 1 + (-5.00 + 7.48i)T + (-8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (3.13 - 0.308i)T + (28.4 - 5.65i)T^{2} \) |
| 31 | \( 1 + (-7.20 - 7.20i)T + 31iT^{2} \) |
| 37 | \( 1 + (9.09 + 4.86i)T + (20.5 + 30.7i)T^{2} \) |
| 41 | \( 1 + (-1.24 - 0.831i)T + (15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (5.60 - 4.59i)T + (8.38 - 42.1i)T^{2} \) |
| 47 | \( 1 + (1.81 - 0.749i)T + (33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (0.426 + 0.0419i)T + (51.9 + 10.3i)T^{2} \) |
| 59 | \( 1 + (-3.61 + 1.09i)T + (49.0 - 32.7i)T^{2} \) |
| 61 | \( 1 + (-5.20 + 6.34i)T + (-11.9 - 59.8i)T^{2} \) |
| 67 | \( 1 + (-4.89 + 5.95i)T + (-13.0 - 65.7i)T^{2} \) |
| 71 | \( 1 + (0.384 - 1.93i)T + (-65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (15.7 - 3.13i)T + (67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (-3.62 - 1.50i)T + (55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (10.5 - 5.63i)T + (46.1 - 69.0i)T^{2} \) |
| 89 | \( 1 + (2.45 + 3.67i)T + (-34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (-5.30 - 5.30i)T + 97iT^{2} \) |
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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
−10.23291111550318244366581698511, −9.276263264387031093978624636598, −8.623451746646728483432941802280, −8.136622236585729554610818737729, −6.66747081944534522431785688778, −5.08358300444134082915632965357, −4.51693163596888673290750859122, −3.69188496615410017376648584181, −2.65424974498040614460782943660, −1.47930179890654085160500941076,
1.08022612562267453067640584450, 2.99747651853545916446537834036, 3.51801123272023829529745789566, 5.10823272150750230780474432804, 5.93071521749240874088405757410, 7.07180519052003171592437236479, 7.46292875688055442043006276206, 8.529287236079288771978143273412, 8.672531201216291584715792034481, 9.845041060175773449939341836116
