| L(s) = 1 | + (−0.140 + 1.40i)2-s + (−1.96 − 0.395i)4-s + (2.65 + 0.467i)5-s + (−0.448 − 0.534i)7-s + (0.831 − 2.70i)8-s + (−1.03 + 3.66i)10-s + (1.10 + 6.24i)11-s + (1.64 − 0.598i)13-s + (0.814 − 0.555i)14-s + (3.68 + 1.54i)16-s + (4.12 + 2.38i)17-s + (−0.795 + 0.459i)19-s + (−5.01 − 1.96i)20-s + (−8.94 + 0.672i)22-s + (−4.88 − 4.09i)23-s + ⋯ |
| L(s) = 1 | + (−0.0992 + 0.995i)2-s + (−0.980 − 0.197i)4-s + (1.18 + 0.209i)5-s + (−0.169 − 0.201i)7-s + (0.293 − 0.955i)8-s + (−0.325 + 1.15i)10-s + (0.331 + 1.88i)11-s + (0.455 − 0.165i)13-s + (0.217 − 0.148i)14-s + (0.921 + 0.387i)16-s + (1.00 + 0.578i)17-s + (−0.182 + 0.105i)19-s + (−1.12 − 0.439i)20-s + (−1.90 + 0.143i)22-s + (−1.01 − 0.853i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0916 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0916 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.921695 + 1.01039i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.921695 + 1.01039i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.140 - 1.40i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-2.65 - 0.467i)T + (4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (0.448 + 0.534i)T + (-1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (-1.10 - 6.24i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (-1.64 + 0.598i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-4.12 - 2.38i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.795 - 0.459i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.88 + 4.09i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (1.69 - 4.66i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-1.35 + 1.61i)T + (-5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (0.943 - 1.63i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.208 + 0.571i)T + (-31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.489 + 0.0863i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-5.04 + 4.23i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + 5.84iT - 53T^{2} \) |
| 59 | \( 1 + (-0.785 + 4.45i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-3.35 + 2.81i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (0.703 + 1.93i)T + (-51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (3.99 - 6.91i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (5.46 + 9.47i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.23 + 14.3i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-11.0 - 4.01i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (8.08 - 4.67i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.0861 - 0.488i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−12.20161259057324073120552721236, −10.26304824068398867428419715929, −10.06706106562911088117721663564, −9.084065293617447235434703542427, −7.933570005913450427551188091564, −6.89827245359820705442673997366, −6.14590625926517163255053119266, −5.13405082245112236279283343574, −3.91095255187931626087758538341, −1.82898698233317428408082558393,
1.20469591825155060231756589767, 2.71984712330621338891430948912, 3.86829302229138827746493459658, 5.54180196844215006677487879230, 6.00984397756161323980956523581, 7.909454965014336343690111578416, 8.926804541038941755848066168296, 9.534680240333323420539592280573, 10.44473220685758844679007165291, 11.36964427662327448853206462305
