| L(s) = 1 | + (0.261 − 0.0124i)2-s + (−1.92 + 0.183i)4-s + (3.04 + 2.90i)5-s + (−0.702 − 3.64i)7-s + (−1.01 + 0.146i)8-s + (0.832 + 0.721i)10-s + (3.45 − 1.78i)11-s + (0.259 + 0.0499i)13-s + (−0.229 − 0.944i)14-s + (3.52 − 0.680i)16-s + (1.22 + 2.67i)17-s + (1.48 + 0.679i)19-s + (−6.38 − 5.02i)20-s + (0.882 − 0.509i)22-s + (3.98 + 2.67i)23-s + ⋯ |
| L(s) = 1 | + (0.184 − 0.00881i)2-s + (−0.961 + 0.0917i)4-s + (1.36 + 1.29i)5-s + (−0.265 − 1.37i)7-s + (−0.360 + 0.0518i)8-s + (0.263 + 0.228i)10-s + (1.04 − 0.537i)11-s + (0.0718 + 0.0138i)13-s + (−0.0612 − 0.252i)14-s + (0.882 − 0.170i)16-s + (0.295 + 0.647i)17-s + (0.341 + 0.155i)19-s + (−1.42 − 1.12i)20-s + (0.188 − 0.108i)22-s + (0.830 + 0.557i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 621 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 - 0.371i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 621 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.928 - 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.67145 + 0.322024i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.67145 + 0.322024i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + (-3.98 - 2.67i)T \) |
| good | 2 | \( 1 + (-0.261 + 0.0124i)T + (1.99 - 0.190i)T^{2} \) |
| 5 | \( 1 + (-3.04 - 2.90i)T + (0.237 + 4.99i)T^{2} \) |
| 7 | \( 1 + (0.702 + 3.64i)T + (-6.49 + 2.60i)T^{2} \) |
| 11 | \( 1 + (-3.45 + 1.78i)T + (6.38 - 8.96i)T^{2} \) |
| 13 | \( 1 + (-0.259 - 0.0499i)T + (12.0 + 4.83i)T^{2} \) |
| 17 | \( 1 + (-1.22 - 2.67i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-1.48 - 0.679i)T + (12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (0.234 - 2.45i)T + (-28.4 - 5.48i)T^{2} \) |
| 31 | \( 1 + (-0.601 + 0.472i)T + (7.30 - 30.1i)T^{2} \) |
| 37 | \( 1 + (-1.05 + 3.58i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (2.75 - 2.89i)T + (-1.95 - 40.9i)T^{2} \) |
| 43 | \( 1 + (4.40 - 5.60i)T + (-10.1 - 41.7i)T^{2} \) |
| 47 | \( 1 + (-4.75 - 2.74i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.42 + 1.64i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-1.82 + 9.49i)T + (-54.7 - 21.9i)T^{2} \) |
| 61 | \( 1 + (-1.51 + 3.79i)T + (-44.1 - 42.0i)T^{2} \) |
| 67 | \( 1 + (-6.13 + 11.9i)T + (-38.8 - 54.5i)T^{2} \) |
| 71 | \( 1 + (-0.327 - 0.510i)T + (-29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-6.89 + 15.0i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (-1.99 + 0.691i)T + (62.0 - 48.8i)T^{2} \) |
| 83 | \( 1 + (8.75 - 8.34i)T + (3.94 - 82.9i)T^{2} \) |
| 89 | \( 1 + (1.15 - 8.01i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (5.87 + 1.42i)T + (86.2 + 44.4i)T^{2} \) |
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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
−10.61081492534312862216720459485, −9.702326570358414522788626781887, −9.338853227444390767908427384748, −7.975452266437657462156083328870, −6.86873371411509278798136058244, −6.26299708204997967781276870523, −5.23118799790531491026897453624, −3.81951802276506496459476766746, −3.20859705199412550317326890523, −1.33513634070355093034605885787,
1.18297707571931333119790380506, 2.57136014154776797184431723811, 4.25058601116584765276200159851, 5.25639907275149369126276841838, 5.62075303424255214626476164328, 6.70625677717510945644180320533, 8.511968973529420987085168788776, 8.909486144760468902111254900684, 9.524242378739527918559528627655, 10.07014380148700392351164137376
