| L(s) = 1 | + (−0.392 − 0.827i)2-s + (0.706 − 0.651i)3-s + (0.733 − 0.898i)4-s + (0.686 − 2.12i)5-s + (−0.816 − 0.328i)6-s + (1.37 − 0.396i)7-s + (−2.81 − 0.693i)8-s + (−0.167 + 2.07i)9-s + (−2.03 + 0.267i)10-s + (−0.827 + 0.973i)11-s + (−0.0672 − 1.11i)12-s + (2.70 − 2.38i)13-s + (−0.866 − 0.978i)14-s + (−0.901 − 1.94i)15-s + (0.0666 + 0.326i)16-s + (1.13 − 2.06i)17-s + ⋯ |
| L(s) = 1 | + (−0.277 − 0.585i)2-s + (0.407 − 0.376i)3-s + (0.366 − 0.449i)4-s + (0.306 − 0.951i)5-s + (−0.333 − 0.134i)6-s + (0.517 − 0.149i)7-s + (−0.994 − 0.245i)8-s + (−0.0557 + 0.690i)9-s + (−0.642 + 0.0847i)10-s + (−0.249 + 0.293i)11-s + (−0.0194 − 0.321i)12-s + (0.748 − 0.662i)13-s + (−0.231 − 0.261i)14-s + (−0.232 − 0.503i)15-s + (0.0166 + 0.0815i)16-s + (0.275 − 0.500i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.793 + 0.608i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.793 + 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.572392 - 1.68598i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.572392 - 1.68598i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.686 + 2.12i)T \) |
| 13 | \( 1 + (-2.70 + 2.38i)T \) |
| good | 2 | \( 1 + (0.392 + 0.827i)T + (-1.26 + 1.54i)T^{2} \) |
| 3 | \( 1 + (-0.706 + 0.651i)T + (0.241 - 2.99i)T^{2} \) |
| 7 | \( 1 + (-1.37 + 0.396i)T + (5.91 - 3.74i)T^{2} \) |
| 11 | \( 1 + (0.827 - 0.973i)T + (-1.76 - 10.8i)T^{2} \) |
| 17 | \( 1 + (-1.13 + 2.06i)T + (-9.08 - 14.3i)T^{2} \) |
| 19 | \( 1 + (0.251 + 0.0673i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (2.02 + 7.55i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (-8.10 + 3.84i)T + (18.3 - 22.4i)T^{2} \) |
| 31 | \( 1 + (8.70 - 6.82i)T + (7.41 - 30.0i)T^{2} \) |
| 37 | \( 1 + (1.37 - 1.83i)T + (-10.2 - 35.5i)T^{2} \) |
| 41 | \( 1 + (-4.60 + 4.25i)T + (3.29 - 40.8i)T^{2} \) |
| 43 | \( 1 + (0.950 - 6.69i)T + (-41.3 - 11.9i)T^{2} \) |
| 47 | \( 1 + (-4.86 - 1.84i)T + (35.1 + 31.1i)T^{2} \) |
| 53 | \( 1 + (-3.17 - 5.24i)T + (-24.6 + 46.9i)T^{2} \) |
| 59 | \( 1 + (1.47 - 0.976i)T + (23.1 - 54.2i)T^{2} \) |
| 61 | \( 1 + (5.21 + 5.43i)T + (-2.45 + 60.9i)T^{2} \) |
| 67 | \( 1 + (-6.05 - 7.42i)T + (-13.4 + 65.6i)T^{2} \) |
| 71 | \( 1 + (1.08 + 4.81i)T + (-64.1 + 30.4i)T^{2} \) |
| 73 | \( 1 + (-0.982 - 1.42i)T + (-25.8 + 68.2i)T^{2} \) |
| 79 | \( 1 + (-4.95 - 1.88i)T + (59.1 + 52.3i)T^{2} \) |
| 83 | \( 1 + (0.468 - 0.892i)T + (-47.1 - 68.3i)T^{2} \) |
| 89 | \( 1 + (15.9 - 4.26i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (-12.1 - 4.06i)T + (77.5 + 58.2i)T^{2} \) |
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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
−10.03858848326051801694857208703, −8.926452464125874279524407725784, −8.372820227494876550249511243233, −7.51070360520280134551401988546, −6.31751109023713820421623709944, −5.38314810510218526256482012092, −4.53442663875842432871688838720, −2.89967881613987345037064480211, −1.94549309986740631335119476580, −0.927315947309341607574716388767,
2.00417052155555598267533397598, 3.25771478172992279603953019147, 3.86182592660221694888714508088, 5.61813158951595799749685586951, 6.29036281137222468719554115834, 7.15051714236191399988780341756, 7.961881391067159175383098598551, 8.792564028940176232858199494425, 9.464447877129547014790040602358, 10.48442278445554060292152498197
