| L(s) = 1 | + (−1.37 + 0.344i)2-s + (0.540 − 0.841i)3-s + (1.76 − 0.945i)4-s + (2.26 + 1.03i)5-s + (−0.451 + 1.34i)6-s + (−2.49 − 0.732i)7-s + (−2.09 + 1.90i)8-s + (−0.415 − 0.909i)9-s + (−3.46 − 0.639i)10-s + (−3.71 − 3.21i)11-s + (0.157 − 1.99i)12-s + (−1.81 − 6.18i)13-s + (3.67 + 0.145i)14-s + (2.09 − 1.34i)15-s + (2.21 − 3.33i)16-s + (0.253 + 1.76i)17-s + ⋯ |
| L(s) = 1 | + (−0.969 + 0.243i)2-s + (0.312 − 0.485i)3-s + (0.881 − 0.472i)4-s + (1.01 + 0.463i)5-s + (−0.184 + 0.547i)6-s + (−0.943 − 0.276i)7-s + (−0.739 + 0.673i)8-s + (−0.138 − 0.303i)9-s + (−1.09 − 0.202i)10-s + (−1.12 − 0.970i)11-s + (0.0455 − 0.575i)12-s + (−0.503 − 1.71i)13-s + (0.982 + 0.0387i)14-s + (0.541 − 0.348i)15-s + (0.553 − 0.832i)16-s + (0.0614 + 0.427i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.221 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.221 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.494443 - 0.619593i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.494443 - 0.619593i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.37 - 0.344i)T \) |
| 3 | \( 1 + (-0.540 + 0.841i)T \) |
| 23 | \( 1 + (-1.87 + 4.41i)T \) |
| good | 5 | \( 1 + (-2.26 - 1.03i)T + (3.27 + 3.77i)T^{2} \) |
| 7 | \( 1 + (2.49 + 0.732i)T + (5.88 + 3.78i)T^{2} \) |
| 11 | \( 1 + (3.71 + 3.21i)T + (1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (1.81 + 6.18i)T + (-10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.253 - 1.76i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (2.93 + 0.422i)T + (18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (-8.21 + 1.18i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (7.54 - 4.85i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (-6.44 + 2.94i)T + (24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (1.79 - 3.93i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-2.68 + 4.17i)T + (-17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 - 8.53T + 47T^{2} \) |
| 53 | \( 1 + (-1.11 + 3.78i)T + (-44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (-4.07 - 13.8i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (5.74 + 8.94i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-7.32 + 6.34i)T + (9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (3.72 + 4.30i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (1.02 - 7.12i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (-0.901 + 0.264i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (5.74 - 2.62i)T + (54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (1.07 + 0.688i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (3.10 - 6.80i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−10.51817364207018384775130304393, −9.754975819891402355423872768470, −8.654668369214715980058991981226, −7.974655071393962874881087295101, −6.96756820733779596069492202524, −6.13514226572401030917659177562, −5.48581137479115805981877577419, −3.04460844598311684250980688967, −2.47800688710412843447341835323, −0.55211679470761672308060713492,
1.93692829223168064860277393265, 2.74843490990535881480829030521, 4.34082512017643599809009765345, 5.59572575987433779029626930271, 6.69122970677133281513696284608, 7.55075057832221743224538320476, 8.806453952392213825587615335913, 9.579665448350346824084371418439, 9.685806388413048014561988695832, 10.68611133151036606799451876683
