| L(s) = 1 | + (5.49 − 3.17i)2-s + (−43.8 + 75.9i)4-s + 425. i·5-s + (1.51e3 + 874. i)7-s + 1.36e3i·8-s + (1.35e3 + 2.34e3i)10-s + (−2.67e3 + 1.54e3i)11-s + (−4.95e3 + 6.17e3i)13-s + 1.11e4·14-s + (−1.27e3 − 2.20e3i)16-s + (1.36e4 − 2.36e4i)17-s + (3.68e4 + 2.12e4i)19-s + (−3.23e4 − 1.86e4i)20-s + (−9.79e3 + 1.69e4i)22-s + (−3.01e4 − 5.22e4i)23-s + ⋯ |
| L(s) = 1 | + (0.485 − 0.280i)2-s + (−0.342 + 0.593i)4-s + 1.52i·5-s + (1.66 + 0.963i)7-s + 0.945i·8-s + (0.427 + 0.740i)10-s + (−0.605 + 0.349i)11-s + (−0.625 + 0.780i)13-s + 1.08·14-s + (−0.0775 − 0.134i)16-s + (0.673 − 1.16i)17-s + (1.23 + 0.711i)19-s + (−0.904 − 0.522i)20-s + (−0.196 + 0.339i)22-s + (−0.516 − 0.894i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.772 - 0.635i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.772 - 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.531728073\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.531728073\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 + (4.95e3 - 6.17e3i)T \) |
| good | 2 | \( 1 + (-5.49 + 3.17i)T + (64 - 110. i)T^{2} \) |
| 5 | \( 1 - 425. iT - 7.81e4T^{2} \) |
| 7 | \( 1 + (-1.51e3 - 874. i)T + (4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (2.67e3 - 1.54e3i)T + (9.74e6 - 1.68e7i)T^{2} \) |
| 17 | \( 1 + (-1.36e4 + 2.36e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-3.68e4 - 2.12e4i)T + (4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (3.01e4 + 5.22e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (1.01e4 + 1.76e4i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 - 4.04e4iT - 2.75e10T^{2} \) |
| 37 | \( 1 + (-2.31e5 + 1.33e5i)T + (4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + (-860. + 496. i)T + (9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (-7.72e4 + 1.33e5i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 - 6.67e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 - 9.55e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + (9.07e5 + 5.23e5i)T + (1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (5.75e5 - 9.96e5i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-7.85e5 + 4.53e5i)T + (3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + (1.04e6 + 6.06e5i)T + (4.54e12 + 7.87e12i)T^{2} \) |
| 73 | \( 1 + 3.26e6iT - 1.10e13T^{2} \) |
| 79 | \( 1 - 5.00e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 4.63e5iT - 2.71e13T^{2} \) |
| 89 | \( 1 + (-1.30e6 + 7.55e5i)T + (2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + (1.12e7 + 6.48e6i)T + (4.03e13 + 6.99e13i)T^{2} \) |
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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
−12.17473186524756996683166132415, −11.74874658447392396148525523766, −10.81534303069315239338811431618, −9.442171555340419046836999929689, −7.990608839937043808007850534195, −7.36136699563302719241236491692, −5.54983268564382250141311555058, −4.53906027099248701800418757438, −2.93214753967601360781828674052, −2.14575636596747222988460864181,
0.68297220932946240349658682969, 1.40658228206739536157852596719, 4.02184575711931850165168842045, 5.07064019348607973917053115396, 5.47902594753085991729582570052, 7.59289040955863821266521491274, 8.309045996946402113845226732354, 9.659100488460789299362081663578, 10.66714104600671334033071408728, 11.91421341874861400923423762207
