| L(s) = 1 | + (−0.982 + 0.473i)2-s + (0.109 + 1.45i)3-s + (−0.504 + 0.633i)4-s + (1.29 − 0.398i)5-s + (−0.796 − 1.37i)6-s + (−0.108 + 0.187i)7-s + (0.682 − 2.98i)8-s + (0.860 − 0.129i)9-s + (−1.08 + 1.00i)10-s + (−3.76 − 4.71i)11-s + (−0.976 − 0.665i)12-s + (2.10 + 1.95i)13-s + (0.0176 − 0.235i)14-s + (0.720 + 1.83i)15-s + (0.383 + 1.68i)16-s + (0.270 + 0.0833i)17-s + ⋯ |
| L(s) = 1 | + (−0.695 + 0.334i)2-s + (0.0629 + 0.840i)3-s + (−0.252 + 0.316i)4-s + (0.577 − 0.178i)5-s + (−0.325 − 0.562i)6-s + (−0.0408 + 0.0708i)7-s + (0.241 − 1.05i)8-s + (0.286 − 0.0432i)9-s + (−0.341 + 0.316i)10-s + (−1.13 − 1.42i)11-s + (−0.281 − 0.192i)12-s + (0.584 + 0.542i)13-s + (0.00471 − 0.0629i)14-s + (0.185 + 0.473i)15-s + (0.0959 + 0.420i)16-s + (0.0654 + 0.0202i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.411 - 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.411 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.499348 + 0.322334i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.499348 + 0.322334i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-5.90 - 2.85i)T \) |
| good | 2 | \( 1 + (0.982 - 0.473i)T + (1.24 - 1.56i)T^{2} \) |
| 3 | \( 1 + (-0.109 - 1.45i)T + (-2.96 + 0.447i)T^{2} \) |
| 5 | \( 1 + (-1.29 + 0.398i)T + (4.13 - 2.81i)T^{2} \) |
| 7 | \( 1 + (0.108 - 0.187i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (3.76 + 4.71i)T + (-2.44 + 10.7i)T^{2} \) |
| 13 | \( 1 + (-2.10 - 1.95i)T + (0.971 + 12.9i)T^{2} \) |
| 17 | \( 1 + (-0.270 - 0.0833i)T + (14.0 + 9.57i)T^{2} \) |
| 19 | \( 1 + (-1.12 - 0.169i)T + (18.1 + 5.60i)T^{2} \) |
| 23 | \( 1 + (1.44 - 3.67i)T + (-16.8 - 15.6i)T^{2} \) |
| 29 | \( 1 + (-0.515 + 6.88i)T + (-28.6 - 4.32i)T^{2} \) |
| 31 | \( 1 + (8.17 + 5.57i)T + (11.3 + 28.8i)T^{2} \) |
| 37 | \( 1 + (3.77 + 6.53i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.62 - 2.22i)T + (25.5 - 32.0i)T^{2} \) |
| 47 | \( 1 + (-0.288 + 0.361i)T + (-10.4 - 45.8i)T^{2} \) |
| 53 | \( 1 + (6.12 - 5.68i)T + (3.96 - 52.8i)T^{2} \) |
| 59 | \( 1 + (-1.85 - 8.14i)T + (-53.1 + 25.5i)T^{2} \) |
| 61 | \( 1 + (-11.0 + 7.56i)T + (22.2 - 56.7i)T^{2} \) |
| 67 | \( 1 + (-6.22 - 0.938i)T + (64.0 + 19.7i)T^{2} \) |
| 71 | \( 1 + (0.540 + 1.37i)T + (-52.0 + 48.2i)T^{2} \) |
| 73 | \( 1 + (-0.601 - 0.557i)T + (5.45 + 72.7i)T^{2} \) |
| 79 | \( 1 + (3.07 - 5.33i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (0.538 + 7.18i)T + (-82.0 + 12.3i)T^{2} \) |
| 89 | \( 1 + (-0.331 - 4.42i)T + (-88.0 + 13.2i)T^{2} \) |
| 97 | \( 1 + (2.78 + 3.49i)T + (-21.5 + 94.5i)T^{2} \) |
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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
−16.17312455438772815520426825272, −15.65323093213403261781778616571, −13.81415839453158211239535836681, −12.99301926223276711372145960414, −11.08886189758593039526850589438, −9.835688905428098912897863751657, −8.996181798707910439650829157117, −7.70460807350004258772833565788, −5.71852402652614412247427193879, −3.80393833045568166050856234869,
1.95501408328108238800113214660, 5.22612001412813298720435669375, 7.04183416608363028040395053365, 8.345007027817108548039303057863, 9.916390160612357466343641284771, 10.58684126308794539867210721150, 12.43606751635015046336128264534, 13.34974710753938436707412617679, 14.47345613565719722999358922732, 15.84895750079935984728481795602
