| L(s) = 1 | + (0.366 − 1.36i)2-s + (0.534 − 2.95i)3-s + (−1.73 − i)4-s + (4.34 − 2.46i)5-s + (−3.83 − 1.81i)6-s + (−0.882 + 3.29i)7-s + (−2 + 1.99i)8-s + (−8.42 − 3.15i)9-s + (−1.77 − 6.84i)10-s + (−0.641 − 1.11i)11-s + (−3.87 + 4.57i)12-s + (−1.05 − 3.94i)13-s + (4.17 + 2.41i)14-s + (−4.95 − 14.1i)15-s + (1.99 + 3.46i)16-s + (19.6 + 19.6i)17-s + ⋯ |
| L(s) = 1 | + (0.183 − 0.683i)2-s + (0.178 − 0.984i)3-s + (−0.433 − 0.250i)4-s + (0.869 − 0.493i)5-s + (−0.639 − 0.301i)6-s + (−0.126 + 0.470i)7-s + (−0.250 + 0.249i)8-s + (−0.936 − 0.350i)9-s + (−0.177 − 0.684i)10-s + (−0.0583 − 0.100i)11-s + (−0.323 + 0.381i)12-s + (−0.0813 − 0.303i)13-s + (0.298 + 0.172i)14-s + (−0.330 − 0.943i)15-s + (0.124 + 0.216i)16-s + (1.15 + 1.15i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.410 + 0.911i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.410 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.836598 - 1.29475i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.836598 - 1.29475i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.366 + 1.36i)T \) |
| 3 | \( 1 + (-0.534 + 2.95i)T \) |
| 5 | \( 1 + (-4.34 + 2.46i)T \) |
| good | 7 | \( 1 + (0.882 - 3.29i)T + (-42.4 - 24.5i)T^{2} \) |
| 11 | \( 1 + (0.641 + 1.11i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (1.05 + 3.94i)T + (-146. + 84.5i)T^{2} \) |
| 17 | \( 1 + (-19.6 - 19.6i)T + 289iT^{2} \) |
| 19 | \( 1 + 17.2iT - 361T^{2} \) |
| 23 | \( 1 + (-7.98 - 29.8i)T + (-458. + 264.5i)T^{2} \) |
| 29 | \( 1 + (-46.0 + 26.6i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (13.3 - 23.1i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-6.39 - 6.39i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 + (24.7 - 42.8i)T + (-840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (44.1 + 11.8i)T + (1.60e3 + 924.5i)T^{2} \) |
| 47 | \( 1 + (-2.93 + 10.9i)T + (-1.91e3 - 1.10e3i)T^{2} \) |
| 53 | \( 1 + (-0.174 + 0.174i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + (88.3 + 51.0i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (26.3 + 45.6i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (57.3 - 15.3i)T + (3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 - 5.54T + 5.04e3T^{2} \) |
| 73 | \( 1 + (48.9 - 48.9i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + (84.7 - 48.9i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-69.6 - 18.6i)T + (5.96e3 + 3.44e3i)T^{2} \) |
| 89 | \( 1 + 46.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (41.3 - 154. i)T + (-8.14e3 - 4.70e3i)T^{2} \) |
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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
−13.32549614212298820191570947325, −12.57916642628150340568745256591, −11.71724459664246679977243167826, −10.25881723037990057783155170153, −9.121136569825716981749564971211, −8.079736943425993494368088523331, −6.32638230871145918968844625014, −5.26387935342672117594740994515, −2.97126982540941525818932674996, −1.41859504540993853855322846145,
3.10219947220392614053104755454, 4.72354339005246214530676360813, 5.92927371637946142570701555339, 7.26915321374298506810649239089, 8.762990858512056312419071815064, 9.879432563988974010444220994627, 10.55905576997997989056351202869, 12.14778854768967834161024241871, 13.70383646532325382131125581689, 14.27163230020761896029340797785
