| L(s) = 1 | + (−2.47 − 7.60i)2-s + (−34.2 + 7.27i)3-s + (−51.7 + 37.6i)4-s + (141. − 245. i)5-s + (139. + 242. i)6-s + (360. + 160. i)7-s + (414. + 300. i)8-s + (−879. + 391. i)9-s + (−2.21e3 − 471. i)10-s + (746. + 7.10e3i)11-s + (1.49e3 − 1.66e3i)12-s + (2.94e3 + 3.27e3i)13-s + (330. − 3.14e3i)14-s + (−3.06e3 + 9.43e3i)15-s + (1.26e3 − 3.89e3i)16-s + (3.03e3 − 2.88e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 − 0.672i)2-s + (−0.731 + 0.155i)3-s + (−0.404 + 0.293i)4-s + (0.507 − 0.878i)5-s + (0.264 + 0.458i)6-s + (0.397 + 0.176i)7-s + (0.286 + 0.207i)8-s + (−0.402 + 0.179i)9-s + (−0.701 − 0.149i)10-s + (0.169 + 1.60i)11-s + (0.250 − 0.277i)12-s + (0.371 + 0.412i)13-s + (0.0321 − 0.305i)14-s + (−0.234 + 0.722i)15-s + (0.0772 − 0.237i)16-s + (0.149 − 1.42i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.274 + 0.961i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.274 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.04984 - 0.792291i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.04984 - 0.792291i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.47 + 7.60i)T \) |
| 31 | \( 1 + (1.37e5 - 9.22e4i)T \) |
| good | 3 | \( 1 + (34.2 - 7.27i)T + (1.99e3 - 889. i)T^{2} \) |
| 5 | \( 1 + (-141. + 245. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 7 | \( 1 + (-360. - 160. i)T + (5.51e5 + 6.12e5i)T^{2} \) |
| 11 | \( 1 + (-746. - 7.10e3i)T + (-1.90e7 + 4.05e6i)T^{2} \) |
| 13 | \( 1 + (-2.94e3 - 3.27e3i)T + (-6.55e6 + 6.24e7i)T^{2} \) |
| 17 | \( 1 + (-3.03e3 + 2.88e4i)T + (-4.01e8 - 8.53e7i)T^{2} \) |
| 19 | \( 1 + (-3.08e4 + 3.42e4i)T + (-9.34e7 - 8.88e8i)T^{2} \) |
| 23 | \( 1 + (-7.55e3 - 5.48e3i)T + (1.05e9 + 3.23e9i)T^{2} \) |
| 29 | \( 1 + (7.64e4 + 2.35e5i)T + (-1.39e10 + 1.01e10i)T^{2} \) |
| 37 | \( 1 + (-7.45e4 - 1.29e5i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 + (-7.10e5 - 1.50e5i)T + (1.77e11 + 7.92e10i)T^{2} \) |
| 43 | \( 1 + (-5.90e5 + 6.55e5i)T + (-2.84e10 - 2.70e11i)T^{2} \) |
| 47 | \( 1 + (-2.58e3 + 7.94e3i)T + (-4.09e11 - 2.97e11i)T^{2} \) |
| 53 | \( 1 + (-9.43e5 + 4.20e5i)T + (7.86e11 - 8.72e11i)T^{2} \) |
| 59 | \( 1 + (-1.97e6 + 4.19e5i)T + (2.27e12 - 1.01e12i)T^{2} \) |
| 61 | \( 1 + 2.98e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + (2.31e5 - 4.00e5i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-2.36e6 + 1.05e6i)T + (6.08e12 - 6.75e12i)T^{2} \) |
| 73 | \( 1 + (-4.95e5 - 4.71e6i)T + (-1.08e13 + 2.29e12i)T^{2} \) |
| 79 | \( 1 + (-3.55e4 + 3.38e5i)T + (-1.87e13 - 3.99e12i)T^{2} \) |
| 83 | \( 1 + (2.80e6 + 5.96e5i)T + (2.47e13 + 1.10e13i)T^{2} \) |
| 89 | \( 1 + (2.36e6 - 1.71e6i)T + (1.36e13 - 4.20e13i)T^{2} \) |
| 97 | \( 1 + (-9.62e6 + 6.99e6i)T + (2.49e13 - 7.68e13i)T^{2} \) |
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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
−13.06878061135978951385081979226, −11.91307686803738251261378295674, −11.28233129137956334620254016331, −9.746168474419296823165096623220, −9.042577306107525031997275771186, −7.34371950305452781799659136691, −5.40683931544628647494381879050, −4.59025181635665953753353984023, −2.26402716737023258147634243160, −0.74982391715108435523184579034,
1.04703793851657076478709813325, 3.42721783511861141480734052250, 5.75584326162716290582097013367, 6.08562558818323346291052687969, 7.67705228017278073209944745332, 8.930144906801743851702296365160, 10.60874827381105245869718034716, 11.12605345605986731983236919487, 12.73179191059156055772690788421, 14.20629839180464461178473488823
