| L(s) = 1 | + (−3.95 − 15.5i)2-s + (142. + 46.2i)3-s + (−224. + 122. i)4-s + (−200. + 592. i)5-s + (154. − 2.38e3i)6-s + 4.17e3i·7-s + (2.78e3 + 3.00e3i)8-s + (1.28e4 + 9.31e3i)9-s + (9.97e3 + 766. i)10-s + (−1.62e4 − 2.24e4i)11-s + (−3.76e4 + 7.05e3i)12-s + (6.70e3 + 4.87e3i)13-s + (6.47e4 − 1.65e4i)14-s + (−5.59e4 + 7.49e4i)15-s + (3.54e4 − 5.50e4i)16-s + (−6.07e3 − 1.87e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.247 − 0.969i)2-s + (1.75 + 0.570i)3-s + (−0.877 + 0.478i)4-s + (−0.320 + 0.947i)5-s + (0.119 − 1.84i)6-s + 1.73i·7-s + (0.680 + 0.732i)8-s + (1.95 + 1.41i)9-s + (0.997 + 0.0766i)10-s + (−1.11 − 1.53i)11-s + (−1.81 + 0.340i)12-s + (0.234 + 0.170i)13-s + (1.68 − 0.429i)14-s + (−1.10 + 1.48i)15-s + (0.541 − 0.840i)16-s + (−0.0727 − 0.223i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.282 - 0.959i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.282 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.27665 + 1.70661i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.27665 + 1.70661i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.95 + 15.5i)T \) |
| 5 | \( 1 + (200. - 592. i)T \) |
| good | 3 | \( 1 + (-142. - 46.2i)T + (5.30e3 + 3.85e3i)T^{2} \) |
| 7 | \( 1 - 4.17e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 + (1.62e4 + 2.24e4i)T + (-6.62e7 + 2.03e8i)T^{2} \) |
| 13 | \( 1 + (-6.70e3 - 4.87e3i)T + (2.52e8 + 7.75e8i)T^{2} \) |
| 17 | \( 1 + (6.07e3 + 1.87e4i)T + (-5.64e9 + 4.10e9i)T^{2} \) |
| 19 | \( 1 + (6.56e3 - 2.13e3i)T + (1.37e10 - 9.98e9i)T^{2} \) |
| 23 | \( 1 + (-9.12e4 - 1.25e5i)T + (-2.41e10 + 7.44e10i)T^{2} \) |
| 29 | \( 1 + (3.31e5 - 1.01e6i)T + (-4.04e11 - 2.94e11i)T^{2} \) |
| 31 | \( 1 + (4.85e5 - 1.57e5i)T + (6.90e11 - 5.01e11i)T^{2} \) |
| 37 | \( 1 + (1.35e6 + 9.84e5i)T + (1.08e12 + 3.34e12i)T^{2} \) |
| 41 | \( 1 + (7.38e5 + 5.36e5i)T + (2.46e12 + 7.59e12i)T^{2} \) |
| 43 | \( 1 + 1.05e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + (-3.21e6 - 1.04e6i)T + (1.92e13 + 1.39e13i)T^{2} \) |
| 53 | \( 1 + (1.93e6 - 5.95e6i)T + (-5.03e13 - 3.65e13i)T^{2} \) |
| 59 | \( 1 + (3.25e6 - 4.47e6i)T + (-4.53e13 - 1.39e14i)T^{2} \) |
| 61 | \( 1 + (1.18e6 - 8.61e5i)T + (5.92e13 - 1.82e14i)T^{2} \) |
| 67 | \( 1 + (-2.33e7 + 7.60e6i)T + (3.28e14 - 2.38e14i)T^{2} \) |
| 71 | \( 1 + (-1.58e7 - 5.15e6i)T + (5.22e14 + 3.79e14i)T^{2} \) |
| 73 | \( 1 + (1.46e7 - 1.06e7i)T + (2.49e14 - 7.66e14i)T^{2} \) |
| 79 | \( 1 + (2.04e7 + 6.63e6i)T + (1.22e15 + 8.91e14i)T^{2} \) |
| 83 | \( 1 + (-3.95e7 + 1.28e7i)T + (1.82e15 - 1.32e15i)T^{2} \) |
| 89 | \( 1 + (-7.58e6 + 5.50e6i)T + (1.21e15 - 3.74e15i)T^{2} \) |
| 97 | \( 1 + (1.66e7 - 5.11e7i)T + (-6.34e15 - 4.60e15i)T^{2} \) |
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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
−12.65314149179374540682447002724, −11.24392963373315518780824321827, −10.42471900199978103492868971049, −9.098435086798261919376684903885, −8.660122154098814491874716865028, −7.67137338786183671558553644143, −5.33946256743049658347113509160, −3.50345433062058645605599622359, −2.94062902404972081720265758929, −2.10295515249520415496929631797,
0.50380282282241984508002874884, 1.75232720444803361077197343889, 3.82526899718761167212232729445, 4.67557588260803771163750206718, 6.96211540376798365956309903818, 7.68139565675892490683578484154, 8.225761520961972622638196540520, 9.489304224352378222211663770518, 10.24569406643151501095927978676, 12.80584329809070890041191931758
