| L(s) = 1 | + (−193. − 111. i)2-s + (1.40e3 + 1.67e3i)3-s + (1.67e4 + 2.89e4i)4-s + (−2.00e4 + 1.15e4i)5-s + (−8.43e4 − 4.80e5i)6-s + (3.61e5 − 6.26e5i)7-s − 3.80e6i·8-s + (−8.38e5 + 4.70e6i)9-s + 5.17e6·10-s + (−2.12e7 − 1.22e7i)11-s + (−2.50e7 + 6.86e7i)12-s + (5.32e7 + 9.21e7i)13-s + (−1.39e8 + 8.06e7i)14-s + (−4.75e7 − 1.73e7i)15-s + (−1.50e8 + 2.61e8i)16-s − 1.48e8i·17-s + ⋯ |
| L(s) = 1 | + (−1.51 − 0.871i)2-s + (0.642 + 0.766i)3-s + (1.02 + 1.76i)4-s + (−0.256 + 0.148i)5-s + (−0.301 − 1.71i)6-s + (0.439 − 0.760i)7-s − 1.81i·8-s + (−0.175 + 0.984i)9-s + 0.517·10-s + (−1.09 − 0.629i)11-s + (−0.699 + 1.91i)12-s + (0.848 + 1.46i)13-s + (−1.32 + 0.765i)14-s + (−0.278 − 0.101i)15-s + (−0.561 + 0.972i)16-s − 0.362i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.641 - 0.767i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (-0.641 - 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{15}{2})\) |
\(\approx\) |
\(0.137428 + 0.294078i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.137428 + 0.294078i\) |
| \(L(8)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.40e3 - 1.67e3i)T \) |
| good | 2 | \( 1 + (193. + 111. i)T + (8.19e3 + 1.41e4i)T^{2} \) |
| 5 | \( 1 + (2.00e4 - 1.15e4i)T + (3.05e9 - 5.28e9i)T^{2} \) |
| 7 | \( 1 + (-3.61e5 + 6.26e5i)T + (-3.39e11 - 5.87e11i)T^{2} \) |
| 11 | \( 1 + (2.12e7 + 1.22e7i)T + (1.89e14 + 3.28e14i)T^{2} \) |
| 13 | \( 1 + (-5.32e7 - 9.21e7i)T + (-1.96e15 + 3.40e15i)T^{2} \) |
| 17 | \( 1 + 1.48e8iT - 1.68e17T^{2} \) |
| 19 | \( 1 + 1.39e9T + 7.99e17T^{2} \) |
| 23 | \( 1 + (4.34e9 - 2.51e9i)T + (5.79e18 - 1.00e19i)T^{2} \) |
| 29 | \( 1 + (1.35e9 + 7.80e8i)T + (1.48e20 + 2.57e20i)T^{2} \) |
| 31 | \( 1 + (3.09e9 + 5.35e9i)T + (-3.78e20 + 6.55e20i)T^{2} \) |
| 37 | \( 1 + 8.97e10T + 9.01e21T^{2} \) |
| 41 | \( 1 + (-4.32e10 + 2.49e10i)T + (1.89e22 - 3.28e22i)T^{2} \) |
| 43 | \( 1 + (-1.16e11 + 2.01e11i)T + (-3.69e22 - 6.39e22i)T^{2} \) |
| 47 | \( 1 + (3.23e11 + 1.86e11i)T + (1.28e23 + 2.22e23i)T^{2} \) |
| 53 | \( 1 - 6.91e11iT - 1.37e24T^{2} \) |
| 59 | \( 1 + (1.64e10 - 9.47e9i)T + (3.09e24 - 5.36e24i)T^{2} \) |
| 61 | \( 1 + (7.88e11 - 1.36e12i)T + (-4.93e24 - 8.55e24i)T^{2} \) |
| 67 | \( 1 + (4.27e12 + 7.40e12i)T + (-1.83e25 + 3.18e25i)T^{2} \) |
| 71 | \( 1 - 1.08e13iT - 8.27e25T^{2} \) |
| 73 | \( 1 - 8.40e12T + 1.22e26T^{2} \) |
| 79 | \( 1 + (-7.09e11 + 1.22e12i)T + (-1.84e26 - 3.19e26i)T^{2} \) |
| 83 | \( 1 + (-1.70e13 - 9.83e12i)T + (3.68e26 + 6.37e26i)T^{2} \) |
| 89 | \( 1 - 1.03e13iT - 1.95e27T^{2} \) |
| 97 | \( 1 + (-4.39e13 + 7.61e13i)T + (-3.26e27 - 5.65e27i)T^{2} \) |
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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
−18.62467596318837330879059210124, −16.98085760128179625082447866316, −15.86225312181515808088307908354, −13.76836614802232380341303170829, −11.29686448520995766468536412488, −10.42851657890671577113370428093, −8.921679449647356900080873675303, −7.76030989903611327070841464531, −3.81008554043971647087478562645, −1.97191260046904079617261798595,
0.20878012256404281284812409027, 2.05060513756931088967426963334, 6.10338065683980812528499686417, 7.951733561503225591222532809137, 8.483147352463588323945676362133, 10.38983485763075336689212599432, 12.68446558061141142631038699768, 14.89374303786589312945726482891, 15.76593599801182179608577445919, 17.76484005763639757375716046135
