| L(s) = 1 | + (−8.05 − 7.94i)2-s + (−46.7 + 2.05i)3-s + (1.90 + 127. i)4-s + (181. + 314. i)5-s + (392. + 354. i)6-s + (196. + 113. i)7-s + (1.00e3 − 1.04e3i)8-s + (2.17e3 − 192. i)9-s + (1.03e3 − 3.97e3i)10-s + (1.48e3 + 859. i)11-s + (−352. − 5.97e3i)12-s + (−8.99e3 + 5.19e3i)13-s + (−682. − 2.47e3i)14-s + (−9.12e3 − 1.43e4i)15-s + (−1.63e4 + 488. i)16-s − 1.37e4i·17-s + ⋯ |
| L(s) = 1 | + (−0.712 − 0.701i)2-s + (−0.999 + 0.0439i)3-s + (0.0149 + 0.999i)4-s + (0.649 + 1.12i)5-s + (0.742 + 0.669i)6-s + (0.216 + 0.124i)7-s + (0.691 − 0.722i)8-s + (0.996 − 0.0878i)9-s + (0.326 − 1.25i)10-s + (0.337 + 0.194i)11-s + (−0.0588 − 0.998i)12-s + (−1.13 + 0.655i)13-s + (−0.0664 − 0.240i)14-s + (−0.698 − 1.09i)15-s + (−0.999 + 0.0298i)16-s − 0.676i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.196 - 0.980i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.196 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.487042 + 0.594193i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.487042 + 0.594193i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (8.05 + 7.94i)T \) |
| 3 | \( 1 + (46.7 - 2.05i)T \) |
| good | 5 | \( 1 + (-181. - 314. i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 7 | \( 1 + (-196. - 113. i)T + (4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-1.48e3 - 859. i)T + (9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + (8.99e3 - 5.19e3i)T + (3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 + 1.37e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 - 4.82e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + (-2.29e4 - 3.97e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (4.17e4 - 7.22e4i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 + (-2.04e5 + 1.18e5i)T + (1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 - 1.94e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 + (5.99e5 - 3.46e5i)T + (9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (1.03e5 - 1.78e5i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 + (6.65e5 - 1.15e6i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 + 1.15e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + (2.58e5 - 1.49e5i)T + (1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-1.89e6 - 1.09e6i)T + (1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-9.73e5 - 1.68e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + 4.85e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.18e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + (6.72e6 + 3.88e6i)T + (9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (1.90e6 + 1.09e6i)T + (1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + 1.84e6iT - 4.42e13T^{2} \) |
| 97 | \( 1 + (3.61e6 - 6.26e6i)T + (-4.03e13 - 6.99e13i)T^{2} \) |
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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
−13.26768907813733439617093098734, −11.76601570074225878201032103001, −11.44736836836553634271028940549, −10.00809090220278060922011250306, −9.608829309717836628052418802070, −7.47089078722772546515984176821, −6.62305710317188544402131956505, −4.87401862375697138238077771618, −2.95251250036427960126704996767, −1.43724407462637197269382022218,
0.42274302533164840961959435056, 1.50497940715465297958805871590, 4.88276397786688521237566486507, 5.53527092096134767530918903463, 6.88065634840555908952589344005, 8.198933562140451583637611762436, 9.536260461443704602161509771661, 10.30437806193927182719356235538, 11.71569419564657790754635182438, 12.82801692626790667391897191469
