| L(s) = 1 | + (−3.29 − 10.8i)2-s + (44.4 − 14.5i)3-s + (−106. + 71.3i)4-s + (211. + 365. i)5-s + (−303. − 433. i)6-s + (−1.25e3 − 722. i)7-s + (1.12e3 + 915. i)8-s + (1.76e3 − 1.29e3i)9-s + (3.26e3 − 3.48e3i)10-s + (1.10e3 + 640. i)11-s + (−3.68e3 + 4.71e3i)12-s + (−9.42e3 + 5.43e3i)13-s + (−3.69e3 + 1.59e4i)14-s + (1.46e4 + 1.31e4i)15-s + (6.20e3 − 1.51e4i)16-s + 3.60e4i·17-s + ⋯ |
| L(s) = 1 | + (−0.291 − 0.956i)2-s + (0.950 − 0.310i)3-s + (−0.830 + 0.557i)4-s + (0.755 + 1.30i)5-s + (−0.573 − 0.818i)6-s + (−1.37 − 0.796i)7-s + (0.775 + 0.631i)8-s + (0.807 − 0.590i)9-s + (1.03 − 1.10i)10-s + (0.251 + 0.144i)11-s + (−0.616 + 0.787i)12-s + (−1.18 + 0.686i)13-s + (−0.359 + 1.55i)14-s + (1.12 + 1.00i)15-s + (0.378 − 0.925i)16-s + 1.77i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.682 - 0.731i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.682 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.35711 + 0.589864i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.35711 + 0.589864i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.29 + 10.8i)T \) |
| 3 | \( 1 + (-44.4 + 14.5i)T \) |
| good | 5 | \( 1 + (-211. - 365. i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 7 | \( 1 + (1.25e3 + 722. i)T + (4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-1.10e3 - 640. i)T + (9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + (9.42e3 - 5.43e3i)T + (3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 - 3.60e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 + 6.54e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + (-3.61e4 - 6.26e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (5.75e4 - 9.97e4i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 + (2.94e4 - 1.69e4i)T + (1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 - 3.10e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 + (-5.51e5 + 3.18e5i)T + (9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (8.39e4 - 1.45e5i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 + (3.56e4 - 6.16e4i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 - 3.11e4T + 1.17e12T^{2} \) |
| 59 | \( 1 + (-2.67e5 + 1.54e5i)T + (1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (1.05e6 + 6.08e5i)T + (1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (1.62e6 + 2.82e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 - 7.90e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.94e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + (-1.88e6 - 1.08e6i)T + (9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (6.17e6 + 3.56e6i)T + (1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 - 6.66e6iT - 4.42e13T^{2} \) |
| 97 | \( 1 + (5.22e6 - 9.05e6i)T + (-4.03e13 - 6.99e13i)T^{2} \) |
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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
−13.28169956007917913778418341633, −12.47365369644651978839141217630, −10.73533319881569238089562163801, −9.925045439740328031341619189856, −9.249259621278652279363589440068, −7.46955558338483316089186712998, −6.55408219243402394107014420936, −3.84680796074309612838501347790, −2.92753635114052479058784701024, −1.71287657525601179593583790227,
0.48914073694497081803223057969, 2.61160604297619810521349537252, 4.65775293601286647139807426478, 5.72817002502299904139193970611, 7.26668309329937248236064398251, 8.723620051666384156410989623378, 9.384539353165096526759571681803, 9.904489071202723667013507828569, 12.60070268842313947735282730505, 13.14107364132903518167378315888
