L(s) = 1 | + (1.37 − 2.37i)2-s − 1.36·3-s + (−2.75 − 4.77i)4-s + (0.370 + 0.641i)5-s + (−1.87 + 3.23i)6-s − 9.63·8-s − 1.13·9-s + 2.03·10-s − 1.36·11-s + (3.76 + 6.51i)12-s + (−0.301 + 3.59i)13-s + (−0.505 − 0.875i)15-s + (−7.68 + 13.3i)16-s + (−2.07 − 3.59i)17-s + (−1.55 + 2.69i)18-s − 7.26·19-s + ⋯ |
L(s) = 1 | + (0.969 − 1.67i)2-s − 0.787·3-s + (−1.37 − 2.38i)4-s + (0.165 + 0.287i)5-s + (−0.763 + 1.32i)6-s − 3.40·8-s − 0.379·9-s + 0.642·10-s − 0.411·11-s + (1.08 + 1.88i)12-s + (−0.0837 + 0.996i)13-s + (−0.130 − 0.226i)15-s + (−1.92 + 3.32i)16-s + (−0.503 − 0.871i)17-s + (−0.367 + 0.636i)18-s − 1.66·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0690 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0690 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.406082 + 0.378940i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.406082 + 0.378940i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (0.301 - 3.59i)T \) |
good | 2 | \( 1 + (-1.37 + 2.37i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + 1.36T + 3T^{2} \) |
| 5 | \( 1 + (-0.370 - 0.641i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + 1.36T + 11T^{2} \) |
| 17 | \( 1 + (2.07 + 3.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + 7.26T + 19T^{2} \) |
| 23 | \( 1 + (-1.16 + 2.02i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.203 - 0.353i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1.38 - 2.40i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.05 + 5.28i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.627 - 1.08i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.870 + 1.50i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (2.92 + 5.07i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2.28 - 3.95i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (5.49 + 9.51i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + 6.52T + 61T^{2} \) |
| 67 | \( 1 + 13.7T + 67T^{2} \) |
| 71 | \( 1 + (-2.40 + 4.17i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.03 + 5.25i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.56 - 7.90i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 11.7T + 83T^{2} \) |
| 89 | \( 1 + (0.880 - 1.52i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.76 + 8.25i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.56157846081933039527358792232, −9.408091596949703513863419327843, −8.657022768119527024547766376814, −6.68118517108898369927513505736, −6.00749219548933119545789280465, −4.91022792125433680349811350199, −4.33644863575091805584124804029, −2.92674361301654891588624847584, −2.04018719304843490286313911032, −0.22834720653696736611055329415,
2.99965305309110452730967924755, 4.32114817777628639813876778560, 5.10659574410804343930949318414, 5.93166218774981047064873079608, 6.38879519069210269946297319581, 7.54055159837041317109066366617, 8.336900303107059189435089451019, 9.022107548092504977592650881576, 10.48366998104307799288565533504, 11.42573656767079329621011058291