| L(s) = 1 | + (4 + 6.92i)2-s + (−31.9 + 55.4i)4-s + (105 − 181. i)5-s + (−508 − 879. i)7-s − 511.·8-s + 1.68e3·10-s + (−546 − 945. i)11-s + (−691 + 1.19e3i)13-s + (4.06e3 − 7.03e3i)14-s + (−2.04e3 − 3.54e3i)16-s + 1.47e4·17-s − 3.99e4·19-s + (6.72e3 + 1.16e4i)20-s + (4.36e3 − 7.56e3i)22-s + (−3.43e4 + 5.95e4i)23-s + ⋯ |
| L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.375 − 0.650i)5-s + (−0.559 − 0.969i)7-s − 0.353·8-s + 0.531·10-s + (−0.123 − 0.214i)11-s + (−0.0872 + 0.151i)13-s + (0.395 − 0.685i)14-s + (−0.125 − 0.216i)16-s + 0.725·17-s − 1.33·19-s + (0.187 + 0.325i)20-s + (0.0874 − 0.151i)22-s + (−0.588 + 1.01i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.7177402795\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7177402795\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-4 - 6.92i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-105 + 181. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 7 | \( 1 + (508 + 879. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (546 + 945. i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + (691 - 1.19e3i)T + (-3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 - 1.47e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.99e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + (3.43e4 - 5.95e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-5.12e4 - 8.88e4i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 + (1.13e5 - 1.97e5i)T + (-1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 - 1.60e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (5.42e3 - 9.38e3i)T + (-9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (-3.15e5 - 5.46e5i)T + (-1.35e11 + 2.35e11i)T^{2} \) |
| 47 | \( 1 + (2.36e5 + 4.09e5i)T + (-2.53e11 + 4.38e11i)T^{2} \) |
| 53 | \( 1 + 1.49e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + (1.32e6 - 2.28e6i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (4.13e5 + 7.16e5i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-6.30e4 + 1.09e5i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + 1.41e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 9.80e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + (-1.78e6 - 3.08e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (2.83e6 + 4.91e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + 1.19e7T + 4.42e13T^{2} \) |
| 97 | \( 1 + (4.34e6 + 7.51e6i)T + (-4.03e13 + 6.99e13i)T^{2} \) |
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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
−12.38590282405699067310956022452, −10.92875447280971535719004613922, −9.861474803056141706691768079576, −8.856416775850293193499677398173, −7.71774133683985805629707949672, −6.68285425174787471801328566547, −5.59776990575650498259692524053, −4.44166331907479062758973240146, −3.28627334425985638000464321709, −1.34819969205301651089558808919,
0.16362937463392281351662987648, 2.10434678322277747658240576645, 2.86361606282305645118690798344, 4.30458204941907521524887790694, 5.78086472440448823956642656773, 6.48516845401651517589623430707, 8.118665624761470450068917685764, 9.349678879799478911701259624787, 10.18899675184351352853608829798, 11.06787318355986964097171174062
