| L(s) = 1 | + (1.30 + 0.665i)2-s + (−0.822 + 0.130i)3-s + (0.0875 + 0.120i)4-s + (−0.233 + 2.22i)5-s + (−1.16 − 0.377i)6-s + (0.659 − 4.16i)7-s + (−0.424 − 2.68i)8-s + (−2.19 + 0.712i)9-s + (−1.78 + 2.74i)10-s + (−0.920 + 3.18i)11-s + (−0.0877 − 0.0877i)12-s + (−0.420 + 0.824i)13-s + (3.63 − 4.99i)14-s + (−0.0979 − 1.85i)15-s + (1.32 − 4.06i)16-s + (0.875 + 1.71i)17-s + ⋯ |
| L(s) = 1 | + (0.923 + 0.470i)2-s + (−0.474 + 0.0751i)3-s + (0.0437 + 0.0602i)4-s + (−0.104 + 0.994i)5-s + (−0.473 − 0.153i)6-s + (0.249 − 1.57i)7-s + (−0.150 − 0.947i)8-s + (−0.731 + 0.237i)9-s + (−0.564 + 0.869i)10-s + (−0.277 + 0.960i)11-s + (−0.0253 − 0.0253i)12-s + (−0.116 + 0.228i)13-s + (0.970 − 1.33i)14-s + (−0.0252 − 0.479i)15-s + (0.330 − 1.01i)16-s + (0.212 + 0.416i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 - 0.482i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.875 - 0.482i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.02006 + 0.262680i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.02006 + 0.262680i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.233 - 2.22i)T \) |
| 11 | \( 1 + (0.920 - 3.18i)T \) |
| good | 2 | \( 1 + (-1.30 - 0.665i)T + (1.17 + 1.61i)T^{2} \) |
| 3 | \( 1 + (0.822 - 0.130i)T + (2.85 - 0.927i)T^{2} \) |
| 7 | \( 1 + (-0.659 + 4.16i)T + (-6.65 - 2.16i)T^{2} \) |
| 13 | \( 1 + (0.420 - 0.824i)T + (-7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (-0.875 - 1.71i)T + (-9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (-4.39 - 3.19i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-1.95 + 1.95i)T - 23iT^{2} \) |
| 29 | \( 1 + (0.810 - 0.588i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-0.131 - 0.403i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (4.87 + 0.771i)T + (35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (-0.339 + 0.467i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (5.05 + 5.05i)T + 43iT^{2} \) |
| 47 | \( 1 + (0.186 + 1.17i)T + (-44.6 + 14.5i)T^{2} \) |
| 53 | \( 1 + (-8.09 - 4.12i)T + (31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (5.47 + 7.53i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-7.40 - 2.40i)T + (49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (3.05 + 3.05i)T + 67iT^{2} \) |
| 71 | \( 1 + (-2.65 + 8.17i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-5.39 - 0.854i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (-0.705 - 2.17i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-1.77 + 0.902i)T + (48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 - 13.9iT - 89T^{2} \) |
| 97 | \( 1 + (-1.50 + 2.96i)T + (-57.0 - 78.4i)T^{2} \) |
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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
−15.09922149233619236586153809870, −14.24550948013309605291675894759, −13.63176426623090742931618503633, −12.14083744266960382832170610998, −10.75982640677870642877114052021, −10.01288863065667166949969980175, −7.57057373316167018024527225063, −6.63912850779153218725852455218, −5.15297678792335216523781939925, −3.74359637156638644456182840419,
2.98425097363403209050426868035, 5.15938250516688489393506567922, 5.64145158594276843712038188428, 8.305302708803490014798144703748, 9.100015532871975858153526619841, 11.46215002484084892703654440548, 11.80631839672516289205194383634, 12.82673055628188539560785855274, 13.84518969417529450072254444607, 15.13361516037120901191449174769
