| L(s) = 1 | + (0.654 − 0.755i)2-s + (−0.841 + 0.540i)3-s + (−0.142 − 0.989i)4-s + (−1.37 − 3.01i)5-s + (−0.142 + 0.989i)6-s + (3.66 − 1.07i)7-s + (−0.841 − 0.540i)8-s + (0.415 − 0.909i)9-s + (−3.18 − 0.934i)10-s + (0.230 + 0.265i)11-s + (0.654 + 0.755i)12-s + (−0.367 − 0.107i)13-s + (1.58 − 3.47i)14-s + (2.79 + 1.79i)15-s + (−0.959 + 0.281i)16-s + (−1.03 + 7.20i)17-s + ⋯ |
| L(s) = 1 | + (0.463 − 0.534i)2-s + (−0.485 + 0.312i)3-s + (−0.0711 − 0.494i)4-s + (−0.616 − 1.34i)5-s + (−0.0580 + 0.404i)6-s + (1.38 − 0.407i)7-s + (−0.297 − 0.191i)8-s + (0.138 − 0.303i)9-s + (−1.00 − 0.295i)10-s + (0.0694 + 0.0801i)11-s + (0.189 + 0.218i)12-s + (−0.101 − 0.0299i)13-s + (0.424 − 0.929i)14-s + (0.720 + 0.463i)15-s + (−0.239 + 0.0704i)16-s + (−0.251 + 1.74i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.206 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.206 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.916880 - 0.743628i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.916880 - 0.743628i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.654 + 0.755i)T \) |
| 3 | \( 1 + (0.841 - 0.540i)T \) |
| 23 | \( 1 + (0.368 + 4.78i)T \) |
| good | 5 | \( 1 + (1.37 + 3.01i)T + (-3.27 + 3.77i)T^{2} \) |
| 7 | \( 1 + (-3.66 + 1.07i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (-0.230 - 0.265i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.367 + 0.107i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (1.03 - 7.20i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.592 - 4.11i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (-0.637 + 4.43i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (-3.65 - 2.35i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (3.04 - 6.65i)T + (-24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (-2.92 - 6.40i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-7.42 + 4.76i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 - 10.6T + 47T^{2} \) |
| 53 | \( 1 + (2.43 - 0.714i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (7.57 + 2.22i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (10.6 + 6.84i)T + (25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (3.67 - 4.23i)T + (-9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (-0.217 + 0.250i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (0.0995 + 0.692i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (15.4 + 4.54i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (1.92 - 4.21i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (7.41 - 4.76i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (4.66 + 10.2i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−12.57387707656113521054153326021, −12.11544115947854532277692797301, −11.06882451973860445828302197261, −10.22590400209672203652035662153, −8.687237139038879853897888958506, −7.923281950679440455757983775472, −5.95276706000946319971892622183, −4.65984535693911616807672888833, −4.17119993175035713068828617827, −1.38683658986579426270822232247,
2.73558540865912307513301764698, 4.53287923932845121023510279312, 5.69280045044240594834130311523, 7.19396386748721153706215133389, 7.50790562894961293263230668060, 9.041370583824080458794448598349, 10.88192742507670475816015718565, 11.41204975782783951094362737995, 12.17367179289271183578981294129, 13.78763384848470697015360789452
