| L(s) = 1 | + (−0.102 − 1.41i)2-s + (2.46 − 0.747i)3-s + (−1.97 + 0.290i)4-s + (0.995 − 0.0980i)5-s + (−1.30 − 3.39i)6-s + (1.01 − 1.51i)7-s + (0.612 + 2.76i)8-s + (3.01 − 2.01i)9-s + (−0.240 − 1.39i)10-s + (−0.474 − 0.888i)11-s + (−4.65 + 2.19i)12-s + (0.145 − 1.47i)13-s + (−2.23 − 1.26i)14-s + (2.37 − 0.985i)15-s + (3.83 − 1.14i)16-s + (−1.80 − 0.748i)17-s + ⋯ |
| L(s) = 1 | + (−0.0726 − 0.997i)2-s + (1.42 − 0.431i)3-s + (−0.989 + 0.145i)4-s + (0.445 − 0.0438i)5-s + (−0.533 − 1.38i)6-s + (0.381 − 0.571i)7-s + (0.216 + 0.976i)8-s + (1.00 − 0.671i)9-s + (−0.0760 − 0.440i)10-s + (−0.143 − 0.267i)11-s + (−1.34 + 0.633i)12-s + (0.0403 − 0.409i)13-s + (−0.597 − 0.339i)14-s + (0.614 − 0.254i)15-s + (0.957 − 0.286i)16-s + (−0.438 − 0.181i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.410 + 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.410 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.22451 - 1.89477i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.22451 - 1.89477i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.102 + 1.41i)T \) |
| 5 | \( 1 + (-0.995 + 0.0980i)T \) |
| good | 3 | \( 1 + (-2.46 + 0.747i)T + (2.49 - 1.66i)T^{2} \) |
| 7 | \( 1 + (-1.01 + 1.51i)T + (-2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (0.474 + 0.888i)T + (-6.11 + 9.14i)T^{2} \) |
| 13 | \( 1 + (-0.145 + 1.47i)T + (-12.7 - 2.53i)T^{2} \) |
| 17 | \( 1 + (1.80 + 0.748i)T + (12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (-1.71 - 2.08i)T + (-3.70 + 18.6i)T^{2} \) |
| 23 | \( 1 + (-0.582 + 2.92i)T + (-21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (-4.23 - 2.26i)T + (16.1 + 24.1i)T^{2} \) |
| 31 | \( 1 + (2.63 - 2.63i)T - 31iT^{2} \) |
| 37 | \( 1 + (8.02 + 6.58i)T + (7.21 + 36.2i)T^{2} \) |
| 41 | \( 1 + (0.328 + 0.0653i)T + (37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (-6.72 - 2.04i)T + (35.7 + 23.8i)T^{2} \) |
| 47 | \( 1 + (-0.188 + 0.454i)T + (-33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (3.37 - 1.80i)T + (29.4 - 44.0i)T^{2} \) |
| 59 | \( 1 + (-1.34 - 13.7i)T + (-57.8 + 11.5i)T^{2} \) |
| 61 | \( 1 + (-0.0299 - 0.0987i)T + (-50.7 + 33.8i)T^{2} \) |
| 67 | \( 1 + (0.208 + 0.688i)T + (-55.7 + 37.2i)T^{2} \) |
| 71 | \( 1 + (3.79 + 2.53i)T + (27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (-2.61 - 3.91i)T + (-27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (-0.590 - 1.42i)T + (-55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (7.63 - 6.26i)T + (16.1 - 81.4i)T^{2} \) |
| 89 | \( 1 + (-2.80 - 14.1i)T + (-82.2 + 34.0i)T^{2} \) |
| 97 | \( 1 + (-10.3 + 10.3i)T - 97iT^{2} \) |
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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
−10.36975087379335210698860276552, −9.307556932232177001665972622593, −8.723457519114387183151484836879, −7.963043337601047433293850293554, −7.11444936564414493364161750779, −5.52716721331490330040100646540, −4.30450808982283872629824741943, −3.26699144731602977524433566484, −2.39116055599320178641877366269, −1.25060781380853561557493994735,
1.98888958115445349005822477188, 3.29776163908898595584077320774, 4.44324813753786927995825940343, 5.34151888884233881286559617122, 6.52058164037804525970820738301, 7.51008074578580277536770301391, 8.350782450513469137772394889580, 8.985419039628764875734516496320, 9.561740043476934968052147281393, 10.37735461629370974798135003700
