| L(s) = 1 | + (0.650 + 1.60i)3-s + (−0.966 + 0.283i)5-s + (1.54 + 1.34i)7-s + (−2.15 + 2.08i)9-s + (−2.68 + 0.788i)11-s + (0.303 + 0.471i)13-s + (−1.08 − 1.36i)15-s + (−2.99 − 0.430i)17-s + (3.48 + 4.02i)19-s + (−1.14 + 3.36i)21-s + (−1.38 + 0.631i)23-s + (−3.35 + 2.15i)25-s + (−4.75 − 2.09i)27-s + 1.01i·29-s + (1.77 − 2.75i)31-s + ⋯ |
| L(s) = 1 | + (0.375 + 0.926i)3-s + (−0.432 + 0.126i)5-s + (0.585 + 0.507i)7-s + (−0.717 + 0.696i)9-s + (−0.810 + 0.237i)11-s + (0.0840 + 0.130i)13-s + (−0.279 − 0.352i)15-s + (−0.725 − 0.104i)17-s + (0.800 + 0.923i)19-s + (−0.250 + 0.733i)21-s + (−0.288 + 0.131i)23-s + (−0.670 + 0.431i)25-s + (−0.915 − 0.403i)27-s + 0.188i·29-s + (0.318 − 0.495i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.804 - 0.593i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.804 - 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.386361 + 1.17484i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.386361 + 1.17484i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.650 - 1.60i)T \) |
| 67 | \( 1 + (-8.16 + 0.572i)T \) |
| good | 5 | \( 1 + (0.966 - 0.283i)T + (4.20 - 2.70i)T^{2} \) |
| 7 | \( 1 + (-1.54 - 1.34i)T + (0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (2.68 - 0.788i)T + (9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (-0.303 - 0.471i)T + (-5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (2.99 + 0.430i)T + (16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (-3.48 - 4.02i)T + (-2.70 + 18.8i)T^{2} \) |
| 23 | \( 1 + (1.38 - 0.631i)T + (15.0 - 17.3i)T^{2} \) |
| 29 | \( 1 - 1.01iT - 29T^{2} \) |
| 31 | \( 1 + (-1.77 + 2.75i)T + (-12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 - 0.414T + 37T^{2} \) |
| 41 | \( 1 + (0.649 - 4.51i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (-0.863 - 0.124i)T + (41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + (1.67 - 0.766i)T + (30.7 - 35.5i)T^{2} \) |
| 53 | \( 1 + (-1.21 - 8.46i)T + (-50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (1.75 - 2.72i)T + (-24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (0.230 - 0.785i)T + (-51.3 - 32.9i)T^{2} \) |
| 71 | \( 1 + (2.98 - 0.428i)T + (68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (3.06 + 0.898i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-0.0461 - 0.0718i)T + (-32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (3.01 + 10.2i)T + (-69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (-8.74 - 3.99i)T + (58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 - 3.47iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.51265007115319352280809709577, −9.774879287084696686423500168638, −8.929540738506769482592381916779, −8.066391643374088638195605263605, −7.51400628152231587899788094790, −5.98571130387061822786582628317, −5.13025889997245545280314093610, −4.27246904251211573661213272683, −3.22778423101184651052353526587, −2.08769963420192530130126117231,
0.56811192148165107796491335235, 2.07378874254754569481287403137, 3.22546526670911220468596324450, 4.44608214067630527024173202830, 5.52053237496872764313045303527, 6.65622523554273829925076067325, 7.46843601553510753185246140085, 8.116364886038608120125585586951, 8.780460213825068311377318399611, 9.898092066388471011779776650951
