| L(s) = 1 | + (0.766 − 0.642i)2-s + (1.72 + 0.196i)3-s + (0.173 − 0.984i)4-s + (0.683 − 3.87i)5-s + (1.44 − 0.955i)6-s + (−1.15 + 2.38i)7-s + (−0.500 − 0.866i)8-s + (2.92 + 0.676i)9-s + (−1.96 − 3.40i)10-s + (0.286 + 1.62i)11-s + (0.492 − 1.66i)12-s + (−0.258 + 1.46i)13-s + (0.644 + 2.56i)14-s + (1.93 − 6.53i)15-s + (−0.939 − 0.342i)16-s + (−3.00 − 5.19i)17-s + ⋯ |
| L(s) = 1 | + (0.541 − 0.454i)2-s + (0.993 + 0.113i)3-s + (0.0868 − 0.492i)4-s + (0.305 − 1.73i)5-s + (0.589 − 0.390i)6-s + (−0.436 + 0.899i)7-s + (−0.176 − 0.306i)8-s + (0.974 + 0.225i)9-s + (−0.622 − 1.07i)10-s + (0.0862 + 0.489i)11-s + (0.142 − 0.479i)12-s + (−0.0718 + 0.407i)13-s + (0.172 + 0.685i)14-s + (0.500 − 1.68i)15-s + (−0.234 − 0.0855i)16-s + (−0.727 − 1.26i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.410 + 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.410 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.03761 - 1.31688i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.03761 - 1.31688i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.766 + 0.642i)T \) |
| 3 | \( 1 + (-1.72 - 0.196i)T \) |
| 7 | \( 1 + (1.15 - 2.38i)T \) |
| good | 5 | \( 1 + (-0.683 + 3.87i)T + (-4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 + (-0.286 - 1.62i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (0.258 - 1.46i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (3.00 + 5.19i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.99 - 3.45i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.26 - 1.90i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-0.387 - 2.19i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (0.174 - 0.990i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 - 11.5T + 37T^{2} \) |
| 41 | \( 1 + (-0.699 + 3.96i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (3.28 - 2.75i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-1.57 - 8.91i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (5.68 - 9.84i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.81 + 0.659i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (0.132 + 0.750i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (10.5 + 8.87i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.162 + 0.282i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 0.342T + 73T^{2} \) |
| 79 | \( 1 + (10.5 - 8.85i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (2.11 + 11.9i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (-3.33 + 5.78i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (4.05 - 3.40i)T + (16.8 - 95.5i)T^{2} \) |
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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
−11.47906886803201312627745604197, −9.935468893799459054626129218113, −9.210097629920242227856916618913, −8.886255725758944440903530294965, −7.63233944121625918514586054148, −6.15652492164456427456673538183, −4.94770392674137002232945136889, −4.30513672134685929270402024348, −2.75755360394814618172601343260, −1.59740493139412097978264825797,
2.44252311808370481665829333003, 3.34690602511908825235074288387, 4.24140481781605514428328025692, 6.21011871088621636176341745768, 6.74883632636140274393921345468, 7.55806183433922679553812187842, 8.536549259398306633898657925857, 9.832824217649269203518349320011, 10.57667732674286651797565238060, 11.33796704566603838647921980170
