L(s) = 1 | + (−0.858 + 22.6i)2-s + (163. − 163. i)3-s + (−510. − 38.8i)4-s + (1.39e3 + 135. i)5-s + (3.55e3 + 3.83e3i)6-s + (−7.88e3 − 7.88e3i)7-s + (1.31e3 − 1.15e4i)8-s − 3.37e4i·9-s + (−4.25e3 + 3.13e4i)10-s − 3.58e4i·11-s + (−8.97e4 + 7.70e4i)12-s + (6.21e4 + 6.21e4i)13-s + (1.85e5 − 1.71e5i)14-s + (2.49e5 − 2.05e5i)15-s + (2.59e5 + 3.96e4i)16-s + (1.77e4 − 1.77e4i)17-s + ⋯ |
L(s) = 1 | + (−0.0379 + 0.999i)2-s + (1.16 − 1.16i)3-s + (−0.997 − 0.0758i)4-s + (0.995 + 0.0968i)5-s + (1.11 + 1.20i)6-s + (−1.24 − 1.24i)7-s + (0.113 − 0.993i)8-s − 1.71i·9-s + (−0.134 + 0.990i)10-s − 0.738i·11-s + (−1.24 + 1.07i)12-s + (0.603 + 0.603i)13-s + (1.28 − 1.19i)14-s + (1.27 − 1.04i)15-s + (0.988 + 0.151i)16-s + (0.0514 − 0.0514i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.747 + 0.664i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.747 + 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(2.10350 - 0.799579i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.10350 - 0.799579i\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.858 - 22.6i)T \) |
| 5 | \( 1 + (-1.39e3 - 135. i)T \) |
good | 3 | \( 1 + (-163. + 163. i)T - 1.96e4iT^{2} \) |
| 7 | \( 1 + (7.88e3 + 7.88e3i)T + 4.03e7iT^{2} \) |
| 11 | \( 1 + 3.58e4iT - 2.35e9T^{2} \) |
| 13 | \( 1 + (-6.21e4 - 6.21e4i)T + 1.06e10iT^{2} \) |
| 17 | \( 1 + (-1.77e4 + 1.77e4i)T - 1.18e11iT^{2} \) |
| 19 | \( 1 - 3.81e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + (-1.41e5 + 1.41e5i)T - 1.80e12iT^{2} \) |
| 29 | \( 1 + 1.44e6iT - 1.45e13T^{2} \) |
| 31 | \( 1 - 4.97e5iT - 2.64e13T^{2} \) |
| 37 | \( 1 + (1.34e7 - 1.34e7i)T - 1.29e14iT^{2} \) |
| 41 | \( 1 - 2.65e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + (2.77e6 - 2.77e6i)T - 5.02e14iT^{2} \) |
| 47 | \( 1 + (-1.64e7 - 1.64e7i)T + 1.11e15iT^{2} \) |
| 53 | \( 1 + (2.01e7 + 2.01e7i)T + 3.29e15iT^{2} \) |
| 59 | \( 1 - 7.03e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.52e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + (9.27e7 + 9.27e7i)T + 2.72e16iT^{2} \) |
| 71 | \( 1 - 4.19e8iT - 4.58e16T^{2} \) |
| 73 | \( 1 + (7.79e7 + 7.79e7i)T + 5.88e16iT^{2} \) |
| 79 | \( 1 + 3.89e5T + 1.19e17T^{2} \) |
| 83 | \( 1 + (-2.65e8 + 2.65e8i)T - 1.86e17iT^{2} \) |
| 89 | \( 1 + 4.33e8iT - 3.50e17T^{2} \) |
| 97 | \( 1 + (-5.75e8 + 5.75e8i)T - 7.60e17iT^{2} \) |
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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
−16.12153551504380796581448623628, −14.22250104925066848048499376161, −13.64298251541006510125420068662, −13.01541269217136085855677006520, −9.914768456648063776003472957309, −8.718708378876080369468213388793, −7.17049414458420301399963099423, −6.30682913752403635926104946860, −3.36980360966244101058956830031, −1.01534834711203043486275870186,
2.31866815231112966897229699954, 3.41720775356439162912415574286, 5.38994706103301350337768536721, 8.790229202572619711475127042814, 9.490991923312710150349640400629, 10.34507916713463641318488047610, 12.52465587715066895058635464609, 13.62106061720724793429127294520, 14.91241756683159934832388007517, 16.08737421725467345894675783949