L(s) = 1 | + (−0.819 − 1.52i)3-s + (−0.206 − 0.769i)5-s + (2.17 + 3.76i)7-s + (−1.65 + 2.50i)9-s + (−1.05 + 3.93i)11-s + (−0.454 − 1.69i)13-s + (−1.00 + 0.945i)15-s + 6.68i·17-s + (0.708 + 0.708i)19-s + (3.96 − 6.40i)21-s + (−3.88 − 2.24i)23-s + (3.78 − 2.18i)25-s + (5.17 + 0.475i)27-s + (−1.06 + 3.98i)29-s + (4.94 + 2.85i)31-s + ⋯ |
L(s) = 1 | + (−0.473 − 0.880i)3-s + (−0.0922 − 0.344i)5-s + (0.822 + 1.42i)7-s + (−0.551 + 0.833i)9-s + (−0.318 + 1.18i)11-s + (−0.126 − 0.470i)13-s + (−0.259 + 0.244i)15-s + 1.62i·17-s + (0.162 + 0.162i)19-s + (0.865 − 1.39i)21-s + (−0.809 − 0.467i)23-s + (0.756 − 0.436i)25-s + (0.995 + 0.0914i)27-s + (−0.198 + 0.740i)29-s + (0.887 + 0.512i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.758 - 0.651i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.758 - 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.06823 + 0.395546i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.06823 + 0.395546i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.819 + 1.52i)T \) |
good | 5 | \( 1 + (0.206 + 0.769i)T + (-4.33 + 2.5i)T^{2} \) |
| 7 | \( 1 + (-2.17 - 3.76i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.05 - 3.93i)T + (-9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (0.454 + 1.69i)T + (-11.2 + 6.5i)T^{2} \) |
| 17 | \( 1 - 6.68iT - 17T^{2} \) |
| 19 | \( 1 + (-0.708 - 0.708i)T + 19iT^{2} \) |
| 23 | \( 1 + (3.88 + 2.24i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.06 - 3.98i)T + (-25.1 - 14.5i)T^{2} \) |
| 31 | \( 1 + (-4.94 - 2.85i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.51 + 1.51i)T + 37iT^{2} \) |
| 41 | \( 1 + (-1.36 + 2.36i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-8.60 - 2.30i)T + (37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (1.23 + 2.13i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.68 + 1.68i)T - 53iT^{2} \) |
| 59 | \( 1 + (-1.00 + 0.269i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (-1.97 - 0.528i)T + (52.8 + 30.5i)T^{2} \) |
| 67 | \( 1 + (8.01 - 2.14i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 8.05iT - 71T^{2} \) |
| 73 | \( 1 - 9.73iT - 73T^{2} \) |
| 79 | \( 1 + (11.9 - 6.91i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (3.05 + 0.817i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 - 1.71T + 89T^{2} \) |
| 97 | \( 1 + (-3.66 - 6.34i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.91955102111201274458714806634, −10.10582765986088715271535133252, −8.675041005912522298505198904305, −8.270554822995692258052280850631, −7.31592068868333938131653196650, −6.14252046146744061958049137417, −5.40727101313057570938857558919, −4.51766387419912522539876925549, −2.53188935382619949844661177419, −1.61318632761979622334414241404,
0.72077525172305391170925194412, 2.98070899438232830364218073408, 4.09580527728201085505572166724, 4.86600027934945941255883121671, 5.93546251778572776410571633179, 7.08297805768192010183599874222, 7.88254237881802924760187282687, 9.028159545186781679846162949212, 9.942848990298332785105305223252, 10.70891290581814349272426559216