| L(s) = 1 | + (−0.415 + 0.909i)2-s + (0.959 + 0.281i)3-s + (−0.654 − 0.755i)4-s + (−0.841 + 0.540i)5-s + (−0.654 + 0.755i)6-s + (0.0123 − 0.0857i)7-s + (0.959 − 0.281i)8-s + (0.841 + 0.540i)9-s + (−0.142 − 0.989i)10-s + (−0.0149 − 0.0326i)11-s + (−0.415 − 0.909i)12-s + (0.472 + 3.28i)13-s + (0.0728 + 0.0468i)14-s + (−0.959 + 0.281i)15-s + (−0.142 + 0.989i)16-s + (0.115 − 0.133i)17-s + ⋯ |
| L(s) = 1 | + (−0.293 + 0.643i)2-s + (0.553 + 0.162i)3-s + (−0.327 − 0.377i)4-s + (−0.376 + 0.241i)5-s + (−0.267 + 0.308i)6-s + (0.00465 − 0.0324i)7-s + (0.339 − 0.0996i)8-s + (0.280 + 0.180i)9-s + (−0.0450 − 0.313i)10-s + (−0.00449 − 0.00983i)11-s + (−0.119 − 0.262i)12-s + (0.131 + 0.912i)13-s + (0.0194 + 0.0125i)14-s + (−0.247 + 0.0727i)15-s + (−0.0355 + 0.247i)16-s + (0.0279 − 0.0322i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.408 - 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.408 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.708827 + 1.09330i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.708827 + 1.09330i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.415 - 0.909i)T \) |
| 3 | \( 1 + (-0.959 - 0.281i)T \) |
| 5 | \( 1 + (0.841 - 0.540i)T \) |
| 23 | \( 1 + (3.25 - 3.51i)T \) |
| good | 7 | \( 1 + (-0.0123 + 0.0857i)T + (-6.71 - 1.97i)T^{2} \) |
| 11 | \( 1 + (0.0149 + 0.0326i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (-0.472 - 3.28i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (-0.115 + 0.133i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (-4.96 - 5.72i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (3.67 - 4.23i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (-0.778 + 0.228i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (-4.18 - 2.68i)T + (15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (-2.07 + 1.33i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (2.62 + 0.769i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 + 7.00T + 47T^{2} \) |
| 53 | \( 1 + (0.706 - 4.91i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (-0.652 - 4.53i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (-0.338 + 0.0993i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (-2.11 + 4.62i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (-4.92 + 10.7i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (5.10 + 5.88i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (1.07 + 7.45i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (-6.46 - 4.15i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (-1.10 - 0.324i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (2.62 - 1.68i)T + (40.2 - 88.2i)T^{2} \) |
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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
−10.51582757139807010423446700352, −9.655888332461791579896266538451, −9.035086291625234501245388076499, −7.960058224002897073136766122751, −7.48925382329377583030787287124, −6.44871957886387245823482120226, −5.43324437400811606393815434452, −4.22876558654273136380658914920, −3.32179844516330396608753509365, −1.64635409842310648012162983536,
0.75451527530822083088474236286, 2.37446820846797678425066874074, 3.35328344108723950116606383985, 4.41165960528290478328253126565, 5.54244417364214100959459687179, 6.93789044488662243266434449831, 7.86369584387004508444268427879, 8.458806628181648999055193217136, 9.419514189641892401377437927308, 10.05729869831559724478709180345
