| L(s) = 1 | + (−164. − 94.7i)2-s + (−2.13e3 − 481. i)3-s + (9.78e3 + 1.69e4i)4-s + (9.37e4 − 5.41e4i)5-s + (3.04e5 + 2.81e5i)6-s + (5.84e5 − 1.01e6i)7-s − 6.02e5i·8-s + (4.31e6 + 2.05e6i)9-s − 2.05e7·10-s + (1.28e7 + 7.43e6i)11-s + (−1.27e7 − 4.08e7i)12-s + (−3.40e7 − 5.89e7i)13-s + (−1.91e8 + 1.10e8i)14-s + (−2.26e8 + 7.02e7i)15-s + (1.03e8 − 1.78e8i)16-s − 1.87e8i·17-s + ⋯ |
| L(s) = 1 | + (−1.28 − 0.740i)2-s + (−0.975 − 0.220i)3-s + (0.597 + 1.03i)4-s + (1.19 − 0.692i)5-s + (1.08 + 1.00i)6-s + (0.709 − 1.22i)7-s − 0.287i·8-s + (0.902 + 0.429i)9-s − 2.05·10-s + (0.660 + 0.381i)11-s + (−0.354 − 1.14i)12-s + (−0.542 − 0.939i)13-s + (−1.82 + 1.05i)14-s + (−1.32 + 0.411i)15-s + (0.384 − 0.665i)16-s − 0.457i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.909 + 0.415i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (-0.909 + 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{15}{2})\) |
\(\approx\) |
\(0.173318 - 0.796742i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.173318 - 0.796742i\) |
| \(L(8)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.13e3 + 481. i)T \) |
| good | 2 | \( 1 + (164. + 94.7i)T + (8.19e3 + 1.41e4i)T^{2} \) |
| 5 | \( 1 + (-9.37e4 + 5.41e4i)T + (3.05e9 - 5.28e9i)T^{2} \) |
| 7 | \( 1 + (-5.84e5 + 1.01e6i)T + (-3.39e11 - 5.87e11i)T^{2} \) |
| 11 | \( 1 + (-1.28e7 - 7.43e6i)T + (1.89e14 + 3.28e14i)T^{2} \) |
| 13 | \( 1 + (3.40e7 + 5.89e7i)T + (-1.96e15 + 3.40e15i)T^{2} \) |
| 17 | \( 1 + 1.87e8iT - 1.68e17T^{2} \) |
| 19 | \( 1 - 1.20e9T + 7.99e17T^{2} \) |
| 23 | \( 1 + (1.17e9 - 6.75e8i)T + (5.79e18 - 1.00e19i)T^{2} \) |
| 29 | \( 1 + (-1.46e10 - 8.43e9i)T + (1.48e20 + 2.57e20i)T^{2} \) |
| 31 | \( 1 + (1.75e10 + 3.04e10i)T + (-3.78e20 + 6.55e20i)T^{2} \) |
| 37 | \( 1 + 9.44e10T + 9.01e21T^{2} \) |
| 41 | \( 1 + (1.71e11 - 9.90e10i)T + (1.89e22 - 3.28e22i)T^{2} \) |
| 43 | \( 1 + (-1.04e11 + 1.81e11i)T + (-3.69e22 - 6.39e22i)T^{2} \) |
| 47 | \( 1 + (-2.57e11 - 1.48e11i)T + (1.28e23 + 2.22e23i)T^{2} \) |
| 53 | \( 1 - 3.41e11iT - 1.37e24T^{2} \) |
| 59 | \( 1 + (-1.82e12 + 1.05e12i)T + (3.09e24 - 5.36e24i)T^{2} \) |
| 61 | \( 1 + (1.55e12 - 2.69e12i)T + (-4.93e24 - 8.55e24i)T^{2} \) |
| 67 | \( 1 + (3.94e12 + 6.82e12i)T + (-1.83e25 + 3.18e25i)T^{2} \) |
| 71 | \( 1 + 1.64e13iT - 8.27e25T^{2} \) |
| 73 | \( 1 + 4.02e12T + 1.22e26T^{2} \) |
| 79 | \( 1 + (6.19e12 - 1.07e13i)T + (-1.84e26 - 3.19e26i)T^{2} \) |
| 83 | \( 1 + (-1.44e13 - 8.36e12i)T + (3.68e26 + 6.37e26i)T^{2} \) |
| 89 | \( 1 - 6.17e13iT - 1.95e27T^{2} \) |
| 97 | \( 1 + (-4.70e13 + 8.14e13i)T + (-3.26e27 - 5.65e27i)T^{2} \) |
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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
−17.44782207832348329770645946394, −16.70793827631160465073724049675, −13.74967720478948450984477609175, −12.03243518074115114464524147712, −10.55675338389429873313787153598, −9.594431323062679004829499795154, −7.50081863244598452419291764486, −5.17508989467434641030430635155, −1.59944783863832517374086897521, −0.71091317946039363362723728889,
1.60539024223006189788973730379, 5.62240332902462017255076306472, 6.79065441461557479083531064623, 9.012036961896594638735127143564, 10.18404896344169560490171706279, 11.82712144900202608319841150038, 14.42653804039268123633156948055, 15.93986207390601808490136491028, 17.28598490501284082161421287043, 17.99143812284647536782918649762
