| L(s) = 1 | + (0.684 + 1.53i)3-s + (−1.53 − 2.66i)5-s + (8.16 + 9.06i)7-s + (4.12 − 4.58i)9-s + (2.64 + 12.4i)11-s + (4.55 + 0.478i)13-s + (3.04 − 4.18i)15-s + (−6.15 + 28.9i)17-s + (−3.70 − 35.2i)19-s + (−8.35 + 18.7i)21-s + (−8.42 + 2.73i)23-s + (7.77 − 13.4i)25-s + (24.2 + 7.89i)27-s + (−17.1 − 23.5i)29-s + (29.9 − 8.04i)31-s + ⋯ |
| L(s) = 1 | + (0.228 + 0.512i)3-s + (−0.307 − 0.532i)5-s + (1.16 + 1.29i)7-s + (0.458 − 0.509i)9-s + (0.240 + 1.12i)11-s + (0.350 + 0.0368i)13-s + (0.202 − 0.279i)15-s + (−0.362 + 1.70i)17-s + (−0.194 − 1.85i)19-s + (−0.397 + 0.893i)21-s + (−0.366 + 0.119i)23-s + (0.310 − 0.538i)25-s + (0.899 + 0.292i)27-s + (−0.590 − 0.812i)29-s + (0.965 − 0.259i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.50124 + 0.559943i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.50124 + 0.559943i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 31 | \( 1 + (-29.9 + 8.04i)T \) |
| good | 3 | \( 1 + (-0.684 - 1.53i)T + (-6.02 + 6.68i)T^{2} \) |
| 5 | \( 1 + (1.53 + 2.66i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (-8.16 - 9.06i)T + (-5.12 + 48.7i)T^{2} \) |
| 11 | \( 1 + (-2.64 - 12.4i)T + (-110. + 49.2i)T^{2} \) |
| 13 | \( 1 + (-4.55 - 0.478i)T + (165. + 35.1i)T^{2} \) |
| 17 | \( 1 + (6.15 - 28.9i)T + (-264. - 117. i)T^{2} \) |
| 19 | \( 1 + (3.70 + 35.2i)T + (-353. + 75.0i)T^{2} \) |
| 23 | \( 1 + (8.42 - 2.73i)T + (427. - 310. i)T^{2} \) |
| 29 | \( 1 + (17.1 + 23.5i)T + (-259. + 799. i)T^{2} \) |
| 37 | \( 1 + (31.4 + 18.1i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (38.5 + 17.1i)T + (1.12e3 + 1.24e3i)T^{2} \) |
| 43 | \( 1 + (28.7 - 3.01i)T + (1.80e3 - 384. i)T^{2} \) |
| 47 | \( 1 + (59.0 + 42.8i)T + (682. + 2.10e3i)T^{2} \) |
| 53 | \( 1 + (-9.55 - 8.59i)T + (293. + 2.79e3i)T^{2} \) |
| 59 | \( 1 + (-63.5 + 28.2i)T + (2.32e3 - 2.58e3i)T^{2} \) |
| 61 | \( 1 + 63.3iT - 3.72e3T^{2} \) |
| 67 | \( 1 + (-41.9 - 72.6i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + (45.4 - 50.4i)T + (-526. - 5.01e3i)T^{2} \) |
| 73 | \( 1 + (-13.5 - 63.9i)T + (-4.86e3 + 2.16e3i)T^{2} \) |
| 79 | \( 1 + (-10.5 + 49.6i)T + (-5.70e3 - 2.53e3i)T^{2} \) |
| 83 | \( 1 + (-15.0 + 33.7i)T + (-4.60e3 - 5.11e3i)T^{2} \) |
| 89 | \( 1 + (-71.6 - 23.2i)T + (6.40e3 + 4.65e3i)T^{2} \) |
| 97 | \( 1 + (15.9 - 48.9i)T + (-7.61e3 - 5.53e3i)T^{2} \) |
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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
−13.09425419479546981928510716727, −12.20640134479704866376713221353, −11.38196772302356707222065189977, −10.04069878622374991675183347216, −8.868602089538489737842732530181, −8.313929906708404104585959589148, −6.64280654543174531081651911738, −5.04599869448055848870358599959, −4.14632698017199573707105426405, −1.98346400526487946214431347446,
1.38929733942335892223376644733, 3.48918723204363946468062988142, 4.91895706489549304824288079126, 6.72608515862289895655517698059, 7.65053109070201239836409105862, 8.401945533842036124355902524107, 10.20193520258330184649985526715, 11.00938777540934918829421525868, 11.84485373761321274542768285452, 13.43105811254377500344779078072
