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
−16.08737421725467345894675783949, −14.91241756683159934832388007517, −13.62106061720724793429127294520, −12.52465587715066895058635464609, −10.34507916713463641318488047610, −9.490991923312710150349640400629, −8.790229202572619711475127042814, −5.38994706103301350337768536721, −3.41720775356439162912415574286, −2.31866815231112966897229699954,
1.01534834711203043486275870186, 3.36980360966244101058956830031, 6.30682913752403635926104946860, 7.17049414458420301399963099423, 8.718708378876080369468213388793, 9.914768456648063776003472957309, 13.01541269217136085855677006520, 13.64298251541006510125420068662, 14.22250104925066848048499376161, 16.12153551504380796581448623628