| L(s) = 1 | + (−0.128 − 2.44i)2-s + (−2.33 − 1.88i)3-s + (−3.99 + 0.419i)4-s + (0.975 − 2.01i)5-s + (−4.32 + 5.95i)6-s + (1.49 − 2.18i)7-s + (0.773 + 4.88i)8-s + (1.24 + 5.86i)9-s + (−5.05 − 2.13i)10-s + (3.31 + 0.704i)11-s + (10.1 + 6.56i)12-s + (0.306 + 0.601i)13-s + (−5.53 − 3.38i)14-s + (−6.07 + 2.84i)15-s + (4.00 − 0.850i)16-s + (2.65 + 1.02i)17-s + ⋯ |
| L(s) = 1 | + (−0.0907 − 1.73i)2-s + (−1.34 − 1.08i)3-s + (−1.99 + 0.209i)4-s + (0.436 − 0.899i)5-s + (−1.76 + 2.42i)6-s + (0.565 − 0.824i)7-s + (0.273 + 1.72i)8-s + (0.415 + 1.95i)9-s + (−1.59 − 0.673i)10-s + (0.998 + 0.212i)11-s + (2.91 + 1.89i)12-s + (0.0850 + 0.166i)13-s + (−1.47 − 0.904i)14-s + (−1.56 + 0.735i)15-s + (1.00 − 0.212i)16-s + (0.644 + 0.247i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.449 - 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.449 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.379103 + 0.615251i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.379103 + 0.615251i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.975 + 2.01i)T \) |
| 7 | \( 1 + (-1.49 + 2.18i)T \) |
| good | 2 | \( 1 + (0.128 + 2.44i)T + (-1.98 + 0.209i)T^{2} \) |
| 3 | \( 1 + (2.33 + 1.88i)T + (0.623 + 2.93i)T^{2} \) |
| 11 | \( 1 + (-3.31 - 0.704i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (-0.306 - 0.601i)T + (-7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (-2.65 - 1.02i)T + (12.6 + 11.3i)T^{2} \) |
| 19 | \( 1 + (-0.332 + 3.16i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (5.31 - 0.278i)T + (22.8 - 2.40i)T^{2} \) |
| 29 | \( 1 + (-5.40 - 7.44i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (2.24 + 5.04i)T + (-20.7 + 23.0i)T^{2} \) |
| 37 | \( 1 + (0.910 - 1.40i)T + (-15.0 - 33.8i)T^{2} \) |
| 41 | \( 1 + (0.985 + 0.320i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (-3.97 - 3.97i)T + 43iT^{2} \) |
| 47 | \( 1 + (0.987 + 2.57i)T + (-34.9 + 31.4i)T^{2} \) |
| 53 | \( 1 + (2.51 - 3.11i)T + (-11.0 - 51.8i)T^{2} \) |
| 59 | \( 1 + (-0.797 - 0.886i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (-4.35 - 3.91i)T + (6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (-2.67 + 6.97i)T + (-49.7 - 44.8i)T^{2} \) |
| 71 | \( 1 + (-7.96 + 5.78i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-4.33 + 2.81i)T + (29.6 - 66.6i)T^{2} \) |
| 79 | \( 1 + (1.61 - 3.63i)T + (-52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (-7.87 + 1.24i)T + (78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (8.29 - 9.20i)T + (-9.30 - 88.5i)T^{2} \) |
| 97 | \( 1 + (16.7 + 2.65i)T + (92.2 + 29.9i)T^{2} \) |
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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
−12.05932020569543749484586340825, −11.29454802104053253256145896220, −10.46436594237231434646633882742, −9.410996053071472462920535536736, −8.081911438822248057250288491072, −6.64116768962000445061798648151, −5.22267728761697779222469117363, −4.19078177674714181641790067722, −1.75920235571353308255110760674, −0.916040093939332131301938623055,
4.02149123441931391068838438250, 5.33796676293811056171043531152, 5.94052353117365509674170971048, 6.68428874378760162103752900780, 8.142006091113386967476631306082, 9.403951133770373403550392168532, 10.13972262300077388797229253483, 11.35290023065781604933145310222, 12.18653223990226865756971247544, 14.10905086967439050735492448966
