| L(s) = 1 | + (−0.212 + 0.977i)2-s + (2.04 − 1.53i)3-s + (−0.909 − 0.415i)4-s + (−2.20 − 0.393i)5-s + (1.06 + 2.32i)6-s + (0.361 − 5.04i)7-s + (0.599 − 0.800i)8-s + (0.993 − 3.38i)9-s + (0.852 − 2.06i)10-s + (−1.44 + 2.24i)11-s + (−2.49 + 0.543i)12-s + (6.05 − 0.433i)13-s + (4.85 + 1.42i)14-s + (−5.10 + 2.56i)15-s + (0.654 + 0.755i)16-s + (−0.522 + 1.40i)17-s + ⋯ |
| L(s) = 1 | + (−0.150 + 0.690i)2-s + (1.18 − 0.884i)3-s + (−0.454 − 0.207i)4-s + (−0.984 − 0.175i)5-s + (0.433 + 0.948i)6-s + (0.136 − 1.90i)7-s + (0.211 − 0.283i)8-s + (0.331 − 1.12i)9-s + (0.269 − 0.653i)10-s + (−0.435 + 0.678i)11-s + (−0.720 + 0.156i)12-s + (1.68 − 0.120i)13-s + (1.29 + 0.381i)14-s + (−1.31 + 0.662i)15-s + (0.163 + 0.188i)16-s + (−0.126 + 0.339i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.745 + 0.666i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.745 + 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.27465 - 0.486550i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.27465 - 0.486550i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.212 - 0.977i)T \) |
| 5 | \( 1 + (2.20 + 0.393i)T \) |
| 23 | \( 1 + (4.59 - 1.38i)T \) |
| good | 3 | \( 1 + (-2.04 + 1.53i)T + (0.845 - 2.87i)T^{2} \) |
| 7 | \( 1 + (-0.361 + 5.04i)T + (-6.92 - 0.996i)T^{2} \) |
| 11 | \( 1 + (1.44 - 2.24i)T + (-4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (-6.05 + 0.433i)T + (12.8 - 1.85i)T^{2} \) |
| 17 | \( 1 + (0.522 - 1.40i)T + (-12.8 - 11.1i)T^{2} \) |
| 19 | \( 1 + (0.914 - 2.00i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (-5.74 + 2.62i)T + (18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.449 - 3.12i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (-1.44 - 2.64i)T + (-20.0 + 31.1i)T^{2} \) |
| 41 | \( 1 + (-3.21 + 0.943i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-0.0653 - 0.0872i)T + (-12.1 + 41.2i)T^{2} \) |
| 47 | \( 1 + (1.68 - 1.68i)T - 47iT^{2} \) |
| 53 | \( 1 + (-10.0 - 0.717i)T + (52.4 + 7.54i)T^{2} \) |
| 59 | \( 1 + (9.39 + 8.13i)T + (8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (2.07 - 0.298i)T + (58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (-3.81 - 0.829i)T + (60.9 + 27.8i)T^{2} \) |
| 71 | \( 1 + (3.32 - 2.13i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (7.56 - 2.82i)T + (55.1 - 47.8i)T^{2} \) |
| 79 | \( 1 + (-0.225 + 0.260i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (2.21 - 1.21i)T + (44.8 - 69.8i)T^{2} \) |
| 89 | \( 1 + (1.49 - 10.4i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-3.80 - 2.07i)T + (52.4 + 81.6i)T^{2} \) |
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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
−12.47458520437385744467990830013, −10.99644909867452280567337750953, −10.07872997899045905601897572328, −8.561030948696464141503752350387, −7.995503550879638501526249248343, −7.39321341153977682742923133631, −6.48060652621976610799905908768, −4.36818324698966319851758076586, −3.55897522217107292739485637256, −1.22605951550877455442464267754,
2.54027026166940410042290934874, 3.35892592715962111372695159571, 4.49249020465048987853914333185, 5.98191412629458704644672670480, 8.063659176858966839057097240246, 8.665384532139608293389742537793, 9.072346952191562730509733072624, 10.43226685330847086369233149059, 11.34330856182118416041944030637, 12.07464309104309763501103333973
