| L(s) = 1 | + (−1.44 + 0.229i)2-s + (−0.377 − 0.741i)3-s + (0.141 − 0.0460i)4-s + (1.49 − 1.66i)5-s + (0.717 + 0.987i)6-s + (−3.75 − 1.91i)7-s + (2.41 − 1.23i)8-s + (1.35 − 1.86i)9-s + (−1.78 + 2.74i)10-s + (−0.0877 − 0.0877i)12-s + (−0.144 − 0.914i)13-s + (5.87 + 1.90i)14-s + (−1.79 − 0.481i)15-s + (−3.45 + 2.51i)16-s + (0.301 − 1.90i)17-s + (−1.53 + 3.01i)18-s + ⋯ |
| L(s) = 1 | + (−1.02 + 0.162i)2-s + (−0.218 − 0.428i)3-s + (0.0708 − 0.0230i)4-s + (0.668 − 0.743i)5-s + (0.292 + 0.403i)6-s + (−1.41 − 0.723i)7-s + (0.854 − 0.435i)8-s + (0.452 − 0.622i)9-s + (−0.564 + 0.869i)10-s + (−0.0253 − 0.0253i)12-s + (−0.0401 − 0.253i)13-s + (1.57 + 0.510i)14-s + (−0.464 − 0.124i)15-s + (−0.864 + 0.628i)16-s + (0.0731 − 0.461i)17-s + (−0.361 + 0.710i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.158i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 + 0.158i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0309388 - 0.388908i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0309388 - 0.388908i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.49 + 1.66i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + (1.44 - 0.229i)T + (1.90 - 0.618i)T^{2} \) |
| 3 | \( 1 + (0.377 + 0.741i)T + (-1.76 + 2.42i)T^{2} \) |
| 7 | \( 1 + (3.75 + 1.91i)T + (4.11 + 5.66i)T^{2} \) |
| 13 | \( 1 + (0.144 + 0.914i)T + (-12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (-0.301 + 1.90i)T + (-16.1 - 5.25i)T^{2} \) |
| 19 | \( 1 + (1.67 - 5.16i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-1.95 + 1.95i)T - 23iT^{2} \) |
| 29 | \( 1 + (-0.309 - 0.952i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (0.342 + 0.249i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (2.23 - 4.39i)T + (-21.7 - 29.9i)T^{2} \) |
| 41 | \( 1 + (-0.549 - 0.178i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (5.05 + 5.05i)T + 43iT^{2} \) |
| 47 | \( 1 + (-1.05 + 0.540i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (8.96 - 1.42i)T + (50.4 - 16.3i)T^{2} \) |
| 59 | \( 1 + (8.85 - 2.87i)T + (47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (4.57 + 6.29i)T + (-18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (3.05 + 3.05i)T + 67iT^{2} \) |
| 71 | \( 1 + (6.95 - 5.04i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-2.47 + 4.86i)T + (-42.9 - 59.0i)T^{2} \) |
| 79 | \( 1 + (1.84 + 1.34i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (1.96 + 0.311i)T + (78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 - 13.9iT - 89T^{2} \) |
| 97 | \( 1 + (-0.520 - 3.28i)T + (-92.2 + 29.9i)T^{2} \) |
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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
−9.948993883207684246221098891879, −9.482284514247297195730181359886, −8.649513412778827591268691005917, −7.60717816500926335727270370330, −6.74752545019613188582419760691, −6.05608726499612107566418327839, −4.60122424838375544240348204619, −3.45617244187874203962560825890, −1.47756225192500266586878194285, −0.32495383007216874998568999251,
1.94707879060497367545319569397, 3.10411217779701215855955703185, 4.62692623749042359862630935601, 5.73933431496626486995546695235, 6.68026436898892604386884990247, 7.56878073252689085609825734998, 8.909083714425366265869792334242, 9.407580623153214042200324860015, 10.09106504512549852463973252164, 10.68649594428831059782290674320
