| L(s) = 1 | + (−0.145 + 0.369i)3-s + (−0.0922 − 1.23i)5-s + (0.695 − 1.20i)7-s + (2.08 + 1.93i)9-s + (0.231 − 1.01i)11-s + (−1.47 + 1.00i)13-s + (0.468 + 0.144i)15-s + (0.120 − 1.61i)17-s + (4.89 − 4.53i)19-s + (0.344 + 0.432i)21-s + (−0.790 + 0.243i)23-s + (3.43 − 0.517i)25-s + (−2.09 + 1.00i)27-s + (−0.848 − 2.16i)29-s + (3.28 + 0.494i)31-s + ⋯ |
| L(s) = 1 | + (−0.0837 + 0.213i)3-s + (−0.0412 − 0.550i)5-s + (0.263 − 0.455i)7-s + (0.694 + 0.644i)9-s + (0.0697 − 0.305i)11-s + (−0.409 + 0.279i)13-s + (0.120 + 0.0373i)15-s + (0.0293 − 0.390i)17-s + (1.12 − 1.04i)19-s + (0.0751 + 0.0942i)21-s + (−0.164 + 0.0508i)23-s + (0.687 − 0.103i)25-s + (−0.402 + 0.193i)27-s + (−0.157 − 0.401i)29-s + (0.589 + 0.0888i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.864 + 0.502i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.864 + 0.502i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.51292 - 0.407334i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.51292 - 0.407334i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 43 | \( 1 + (-6.48 + 0.976i)T \) |
| good | 3 | \( 1 + (0.145 - 0.369i)T + (-2.19 - 2.04i)T^{2} \) |
| 5 | \( 1 + (0.0922 + 1.23i)T + (-4.94 + 0.745i)T^{2} \) |
| 7 | \( 1 + (-0.695 + 1.20i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.231 + 1.01i)T + (-9.91 - 4.77i)T^{2} \) |
| 13 | \( 1 + (1.47 - 1.00i)T + (4.74 - 12.1i)T^{2} \) |
| 17 | \( 1 + (-0.120 + 1.61i)T + (-16.8 - 2.53i)T^{2} \) |
| 19 | \( 1 + (-4.89 + 4.53i)T + (1.41 - 18.9i)T^{2} \) |
| 23 | \( 1 + (0.790 - 0.243i)T + (19.0 - 12.9i)T^{2} \) |
| 29 | \( 1 + (0.848 + 2.16i)T + (-21.2 + 19.7i)T^{2} \) |
| 31 | \( 1 + (-3.28 - 0.494i)T + (29.6 + 9.13i)T^{2} \) |
| 37 | \( 1 + (4.10 + 7.10i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-6.37 + 7.98i)T + (-9.12 - 39.9i)T^{2} \) |
| 47 | \( 1 + (-1.95 - 8.55i)T + (-42.3 + 20.3i)T^{2} \) |
| 53 | \( 1 + (-4.54 - 3.09i)T + (19.3 + 49.3i)T^{2} \) |
| 59 | \( 1 + (-1.50 + 0.726i)T + (36.7 - 46.1i)T^{2} \) |
| 61 | \( 1 + (0.126 - 0.0191i)T + (58.2 - 17.9i)T^{2} \) |
| 67 | \( 1 + (3.95 - 3.66i)T + (5.00 - 66.8i)T^{2} \) |
| 71 | \( 1 + (9.81 + 3.02i)T + (58.6 + 39.9i)T^{2} \) |
| 73 | \( 1 + (8.62 - 5.88i)T + (26.6 - 67.9i)T^{2} \) |
| 79 | \( 1 + (-7.86 + 13.6i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (1.25 - 3.20i)T + (-60.8 - 56.4i)T^{2} \) |
| 89 | \( 1 + (4.41 - 11.2i)T + (-65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 + (-0.109 + 0.478i)T + (-87.3 - 42.0i)T^{2} \) |
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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
−10.48470427295925324327020514255, −9.461730003884901659663011027864, −8.851581477626267309337659295297, −7.56331127660390511312108572440, −7.19525752276464764947625611320, −5.71932506593321385622168517896, −4.80963612668040365808370070261, −4.11262941462845559020024414098, −2.57932838885583881350284450770, −1.00457980930475501967099139614,
1.39865706750926124032477240856, 2.84459112451034157551140690938, 3.95216920348326781984327022209, 5.16682823135216833460186658395, 6.17023391440557403215887019599, 7.05869437426948270360986992405, 7.79722016537785928133473529581, 8.836486261338958465651239782074, 9.867099510265033178039632125786, 10.35358349187695779940688232608
