| L(s) = 1 | + (−0.558 + 1.17i)2-s + (1.97 + 1.82i)3-s + (0.193 + 0.236i)4-s + (−2.15 + 0.580i)5-s + (−3.24 + 1.30i)6-s + (2.78 + 0.807i)7-s + (−2.91 + 0.718i)8-s + (0.340 + 4.22i)9-s + (0.522 − 2.86i)10-s + (2.74 + 3.23i)11-s + (−0.0496 + 0.820i)12-s + (3.38 − 1.24i)13-s + (−2.50 + 2.82i)14-s + (−5.32 − 2.79i)15-s + (0.659 − 3.22i)16-s + (1.99 + 3.61i)17-s + ⋯ |
| L(s) = 1 | + (−0.394 + 0.831i)2-s + (1.14 + 1.05i)3-s + (0.0966 + 0.118i)4-s + (−0.965 + 0.259i)5-s + (−1.32 + 0.533i)6-s + (1.05 + 0.305i)7-s + (−1.03 + 0.253i)8-s + (0.113 + 1.40i)9-s + (0.165 − 0.905i)10-s + (0.828 + 0.974i)11-s + (−0.0143 + 0.236i)12-s + (0.938 − 0.344i)13-s + (−0.669 + 0.755i)14-s + (−1.37 − 0.720i)15-s + (0.164 − 0.807i)16-s + (0.482 + 0.876i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.122i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 - 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.119479 + 1.93677i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.119479 + 1.93677i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (2.15 - 0.580i)T \) |
| 13 | \( 1 + (-3.38 + 1.24i)T \) |
| good | 2 | \( 1 + (0.558 - 1.17i)T + (-1.26 - 1.54i)T^{2} \) |
| 3 | \( 1 + (-1.97 - 1.82i)T + (0.241 + 2.99i)T^{2} \) |
| 7 | \( 1 + (-2.78 - 0.807i)T + (5.91 + 3.74i)T^{2} \) |
| 11 | \( 1 + (-2.74 - 3.23i)T + (-1.76 + 10.8i)T^{2} \) |
| 17 | \( 1 + (-1.99 - 3.61i)T + (-9.08 + 14.3i)T^{2} \) |
| 19 | \( 1 + (-0.428 + 0.114i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-1.22 + 4.55i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (3.49 + 1.65i)T + (18.3 + 22.4i)T^{2} \) |
| 31 | \( 1 + (5.83 + 4.57i)T + (7.41 + 30.0i)T^{2} \) |
| 37 | \( 1 + (5.36 + 7.14i)T + (-10.2 + 35.5i)T^{2} \) |
| 41 | \( 1 + (-1.11 - 1.03i)T + (3.29 + 40.8i)T^{2} \) |
| 43 | \( 1 + (0.431 + 3.03i)T + (-41.3 + 11.9i)T^{2} \) |
| 47 | \( 1 + (9.98 - 3.78i)T + (35.1 - 31.1i)T^{2} \) |
| 53 | \( 1 + (-2.49 + 4.12i)T + (-24.6 - 46.9i)T^{2} \) |
| 59 | \( 1 + (-10.2 - 6.75i)T + (23.1 + 54.2i)T^{2} \) |
| 61 | \( 1 + (-6.78 + 7.06i)T + (-2.45 - 60.9i)T^{2} \) |
| 67 | \( 1 + (0.957 - 1.17i)T + (-13.4 - 65.6i)T^{2} \) |
| 71 | \( 1 + (-1.17 + 5.23i)T + (-64.1 - 30.4i)T^{2} \) |
| 73 | \( 1 + (0.642 - 0.931i)T + (-25.8 - 68.2i)T^{2} \) |
| 79 | \( 1 + (-6.60 + 2.50i)T + (59.1 - 52.3i)T^{2} \) |
| 83 | \( 1 + (2.66 + 5.07i)T + (-47.1 + 68.3i)T^{2} \) |
| 89 | \( 1 + (-8.45 - 2.26i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (12.5 - 4.18i)T + (77.5 - 58.2i)T^{2} \) |
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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
−10.48797467787523743799124540731, −9.358647860866459670481739001960, −8.698057284548051296913654053265, −8.133493114199339212480910254373, −7.57175822887227320520850440991, −6.53639528227838000865262046961, −5.22261488130307001707754406143, −4.02773169472806883180823341754, −3.55039890471737298255500551190, −2.16087315716684419260353864078,
1.08407660468005004836525786973, 1.62615571020089937056013491053, 3.17874134217067765704042710114, 3.69313735694780939628882209614, 5.32170754016234453731889972488, 6.71456692898740430228701705814, 7.38860891173583713642958709201, 8.427571749800218876404046421428, 8.678021985174965975717683981592, 9.586659125516441385334164077660
