| L(s) = 1 | + (−0.327 + 0.516i)2-s + (−1.06 + 0.999i)3-s + (0.692 + 1.47i)4-s + (−2.18 − 0.463i)5-s + (−0.167 − 0.876i)6-s + (−0.940 + 0.683i)7-s + (−2.19 − 0.277i)8-s + (−0.0545 + 0.867i)9-s + (0.956 − 0.977i)10-s + (0.878 − 1.38i)11-s + (−2.20 − 0.873i)12-s + (−0.164 + 2.60i)13-s + (−0.0446 − 0.709i)14-s + (2.79 − 1.69i)15-s + (−1.20 + 1.46i)16-s + (−0.117 + 0.250i)17-s + ⋯ |
| L(s) = 1 | + (−0.231 + 0.365i)2-s + (−0.614 + 0.576i)3-s + (0.346 + 0.735i)4-s + (−0.978 − 0.207i)5-s + (−0.0682 − 0.357i)6-s + (−0.355 + 0.258i)7-s + (−0.777 − 0.0982i)8-s + (−0.0181 + 0.289i)9-s + (0.302 − 0.309i)10-s + (0.264 − 0.417i)11-s + (−0.637 − 0.252i)12-s + (−0.0455 + 0.723i)13-s + (−0.0119 − 0.189i)14-s + (0.720 − 0.436i)15-s + (−0.302 + 0.365i)16-s + (−0.0285 + 0.0606i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.779 - 0.626i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.779 - 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.206198 + 0.585668i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.206198 + 0.585668i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (2.18 + 0.463i)T \) |
| good | 2 | \( 1 + (0.327 - 0.516i)T + (-0.851 - 1.80i)T^{2} \) |
| 3 | \( 1 + (1.06 - 0.999i)T + (0.188 - 2.99i)T^{2} \) |
| 7 | \( 1 + (0.940 - 0.683i)T + (2.16 - 6.65i)T^{2} \) |
| 11 | \( 1 + (-0.878 + 1.38i)T + (-4.68 - 9.95i)T^{2} \) |
| 13 | \( 1 + (0.164 - 2.60i)T + (-12.8 - 1.62i)T^{2} \) |
| 17 | \( 1 + (0.117 - 0.250i)T + (-10.8 - 13.0i)T^{2} \) |
| 19 | \( 1 + (-3.66 - 3.44i)T + (1.19 + 18.9i)T^{2} \) |
| 23 | \( 1 + (-4.88 - 1.25i)T + (20.1 + 11.0i)T^{2} \) |
| 29 | \( 1 + (-4.10 - 2.25i)T + (15.5 + 24.4i)T^{2} \) |
| 31 | \( 1 + (-3.63 + 7.73i)T + (-19.7 - 23.8i)T^{2} \) |
| 37 | \( 1 + (1.55 - 1.88i)T + (-6.93 - 36.3i)T^{2} \) |
| 41 | \( 1 + (6.54 - 1.67i)T + (35.9 - 19.7i)T^{2} \) |
| 43 | \( 1 + (1.11 - 3.42i)T + (-34.7 - 25.2i)T^{2} \) |
| 47 | \( 1 + (-8.58 + 1.08i)T + (45.5 - 11.6i)T^{2} \) |
| 53 | \( 1 + (0.991 - 5.19i)T + (-49.2 - 19.5i)T^{2} \) |
| 59 | \( 1 + (10.9 + 4.33i)T + (43.0 + 40.3i)T^{2} \) |
| 61 | \( 1 + (4.28 + 1.09i)T + (53.4 + 29.3i)T^{2} \) |
| 67 | \( 1 + (1.38 - 0.762i)T + (35.9 - 56.5i)T^{2} \) |
| 71 | \( 1 + (-4.95 + 0.625i)T + (68.7 - 17.6i)T^{2} \) |
| 73 | \( 1 + (-11.6 + 4.59i)T + (53.2 - 49.9i)T^{2} \) |
| 79 | \( 1 + (-4.06 + 3.81i)T + (4.96 - 78.8i)T^{2} \) |
| 83 | \( 1 + (6.08 + 5.71i)T + (5.21 + 82.8i)T^{2} \) |
| 89 | \( 1 + (15.6 - 6.18i)T + (64.8 - 60.9i)T^{2} \) |
| 97 | \( 1 + (-13.8 - 7.58i)T + (51.9 + 81.8i)T^{2} \) |
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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
−13.77969520680252270997598308727, −12.42460175294181187921720153596, −11.70708882524373021531824199883, −10.97587340783249185202110369635, −9.479298609780296863878526382647, −8.366061776041545409622071991820, −7.39421679836021547853982162147, −6.13466396024650428492033261151, −4.60108972389967293059605960467, −3.28221999173823328153954861908,
0.78574451394690752848110676216, 3.14852527611014172836539543892, 5.09888069570858276534763344492, 6.57471653554611839519130052449, 7.20787872588860017531293378205, 8.850059714527995532296213063958, 10.12416302221005708519687271311, 11.05348444845598449755642350172, 11.90737957291668089818319494407, 12.56329921764648784373607277513
