| L(s) = 1 | + (0.248 + 0.968i)2-s + (1.56 − 0.298i)3-s + (−0.876 + 0.481i)4-s + (2.23 + 0.132i)5-s + (0.677 + 1.44i)6-s + (2.03 − 2.80i)7-s + (−0.684 − 0.728i)8-s + (−0.434 + 0.171i)9-s + (0.426 + 2.19i)10-s + (−1.85 + 0.475i)11-s + (−1.22 + 1.01i)12-s + (−1.70 − 4.31i)13-s + (3.22 + 1.27i)14-s + (3.52 − 0.458i)15-s + (0.535 − 0.844i)16-s + (−3.55 + 6.47i)17-s + ⋯ |
| L(s) = 1 | + (0.175 + 0.684i)2-s + (0.902 − 0.172i)3-s + (−0.438 + 0.240i)4-s + (0.998 + 0.0592i)5-s + (0.276 + 0.587i)6-s + (0.770 − 1.06i)7-s + (−0.242 − 0.257i)8-s + (−0.144 + 0.0572i)9-s + (0.134 + 0.694i)10-s + (−0.558 + 0.143i)11-s + (−0.354 + 0.292i)12-s + (−0.473 − 1.19i)13-s + (0.861 + 0.341i)14-s + (0.911 − 0.118i)15-s + (0.133 − 0.211i)16-s + (−0.862 + 1.56i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 - 0.534i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.845 - 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.82145 + 0.527312i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.82145 + 0.527312i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.248 - 0.968i)T \) |
| 5 | \( 1 + (-2.23 - 0.132i)T \) |
| good | 3 | \( 1 + (-1.56 + 0.298i)T + (2.78 - 1.10i)T^{2} \) |
| 7 | \( 1 + (-2.03 + 2.80i)T + (-2.16 - 6.65i)T^{2} \) |
| 11 | \( 1 + (1.85 - 0.475i)T + (9.63 - 5.29i)T^{2} \) |
| 13 | \( 1 + (1.70 + 4.31i)T + (-9.47 + 8.89i)T^{2} \) |
| 17 | \( 1 + (3.55 - 6.47i)T + (-9.10 - 14.3i)T^{2} \) |
| 19 | \( 1 + (0.915 - 4.79i)T + (-17.6 - 6.99i)T^{2} \) |
| 23 | \( 1 + (-3.65 - 0.230i)T + (22.8 + 2.88i)T^{2} \) |
| 29 | \( 1 + (7.75 + 0.979i)T + (28.0 + 7.21i)T^{2} \) |
| 31 | \( 1 + (-2.18 - 1.19i)T + (16.6 + 26.1i)T^{2} \) |
| 37 | \( 1 + (0.249 + 0.158i)T + (15.7 + 33.4i)T^{2} \) |
| 41 | \( 1 + (0.502 + 7.99i)T + (-40.6 + 5.13i)T^{2} \) |
| 43 | \( 1 + (2.40 - 0.781i)T + (34.7 - 25.2i)T^{2} \) |
| 47 | \( 1 + (2.87 - 3.06i)T + (-2.95 - 46.9i)T^{2} \) |
| 53 | \( 1 + (-10.3 - 4.85i)T + (33.7 + 40.8i)T^{2} \) |
| 59 | \( 1 + (-2.37 - 2.86i)T + (-11.0 + 57.9i)T^{2} \) |
| 61 | \( 1 + (-0.914 + 14.5i)T + (-60.5 - 7.64i)T^{2} \) |
| 67 | \( 1 + (1.65 + 13.1i)T + (-64.8 + 16.6i)T^{2} \) |
| 71 | \( 1 + (3.22 + 3.02i)T + (4.45 + 70.8i)T^{2} \) |
| 73 | \( 1 + (4.26 + 3.53i)T + (13.6 + 71.7i)T^{2} \) |
| 79 | \( 1 + (0.125 + 0.656i)T + (-73.4 + 29.0i)T^{2} \) |
| 83 | \( 1 + (-12.3 - 2.36i)T + (77.1 + 30.5i)T^{2} \) |
| 89 | \( 1 + (-4.09 + 4.95i)T + (-16.6 - 87.4i)T^{2} \) |
| 97 | \( 1 + (-0.366 + 2.90i)T + (-93.9 - 24.1i)T^{2} \) |
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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
−12.71097346835600250864209654352, −10.82556135017773497514231727266, −10.25992186066745571160100751848, −8.975820724992269162224798998325, −8.059073453171611777313215766375, −7.44573015866612195329882205271, −6.05316419972568099975146520149, −5.03303248748686873838306215149, −3.58458250689597279972527823572, −1.99726714034036413542276367078,
2.18990365875783969522247613008, 2.73104388689747894461869839579, 4.66060519308542253048431208815, 5.48657423055998343509594244048, 7.01043691372475936493987758643, 8.677753011215047290430499182299, 9.046147678548857903921667818305, 9.809027791640924662459400732997, 11.29424072732313793017924161304, 11.70789958574811581637786676315
