| L(s) = 1 | + (−1.91 + 0.263i)2-s + (−1.41 − 0.858i)3-s + (1.68 − 0.471i)4-s + (−0.565 + 1.59i)5-s + (2.93 + 1.27i)6-s + (0.0740 + 1.08i)7-s + (0.448 − 0.194i)8-s + (−0.125 − 0.242i)9-s + (0.664 − 3.19i)10-s + (0.437 + 0.468i)11-s + (−2.77 − 0.778i)12-s + (1.29 − 1.83i)13-s + (−0.427 − 2.05i)14-s + (2.16 − 1.75i)15-s + (−3.79 + 2.30i)16-s + (−0.682 + 0.730i)17-s + ⋯ |
| L(s) = 1 | + (−1.35 + 0.186i)2-s + (−0.814 − 0.495i)3-s + (0.841 − 0.235i)4-s + (−0.252 + 0.711i)5-s + (1.19 + 0.519i)6-s + (0.0279 + 0.409i)7-s + (0.158 − 0.0689i)8-s + (−0.0418 − 0.0808i)9-s + (0.210 − 1.01i)10-s + (0.131 + 0.141i)11-s + (−0.802 − 0.224i)12-s + (0.359 − 0.509i)13-s + (−0.114 − 0.549i)14-s + (0.558 − 0.454i)15-s + (−0.948 + 0.576i)16-s + (−0.165 + 0.177i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 - 0.396i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.918 - 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.417672 + 0.0862276i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.417672 + 0.0862276i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (0.682 - 0.730i)T \) |
| 47 | \( 1 + (2.60 - 6.34i)T \) |
| good | 2 | \( 1 + (1.91 - 0.263i)T + (1.92 - 0.539i)T^{2} \) |
| 3 | \( 1 + (1.41 + 0.858i)T + (1.38 + 2.66i)T^{2} \) |
| 5 | \( 1 + (0.565 - 1.59i)T + (-3.87 - 3.15i)T^{2} \) |
| 7 | \( 1 + (-0.0740 - 1.08i)T + (-6.93 + 0.953i)T^{2} \) |
| 11 | \( 1 + (-0.437 - 0.468i)T + (-0.750 + 10.9i)T^{2} \) |
| 13 | \( 1 + (-1.29 + 1.83i)T + (-4.35 - 12.2i)T^{2} \) |
| 19 | \( 1 + (1.55 + 4.37i)T + (-14.7 + 11.9i)T^{2} \) |
| 23 | \( 1 + (4.83 + 0.664i)T + (22.1 + 6.20i)T^{2} \) |
| 29 | \( 1 + (-1.32 - 1.87i)T + (-9.71 + 27.3i)T^{2} \) |
| 31 | \( 1 + (3.57 - 2.17i)T + (14.2 - 27.5i)T^{2} \) |
| 37 | \( 1 + (-1.32 + 6.39i)T + (-33.9 - 14.7i)T^{2} \) |
| 41 | \( 1 + (7.65 + 3.32i)T + (27.9 + 29.9i)T^{2} \) |
| 43 | \( 1 + (-7.58 + 2.12i)T + (36.7 - 22.3i)T^{2} \) |
| 53 | \( 1 + (-11.1 - 4.85i)T + (36.1 + 38.7i)T^{2} \) |
| 59 | \( 1 + (-6.03 - 1.69i)T + (50.4 + 30.6i)T^{2} \) |
| 61 | \( 1 + (1.44 + 6.96i)T + (-55.9 + 24.3i)T^{2} \) |
| 67 | \( 1 + (0.425 - 6.22i)T + (-66.3 - 9.12i)T^{2} \) |
| 71 | \( 1 + (-11.4 - 1.57i)T + (68.3 + 19.1i)T^{2} \) |
| 73 | \( 1 + (-1.06 + 2.04i)T + (-42.0 - 59.6i)T^{2} \) |
| 79 | \( 1 + (-7.69 + 6.26i)T + (16.0 - 77.3i)T^{2} \) |
| 83 | \( 1 + (-11.4 - 12.2i)T + (-5.66 + 82.8i)T^{2} \) |
| 89 | \( 1 + (-2.59 + 7.28i)T + (-69.0 - 56.1i)T^{2} \) |
| 97 | \( 1 + (-2.28 - 1.38i)T + (44.6 + 86.1i)T^{2} \) |
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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
−10.56200142600956098412921211198, −9.274436978626483864024258037365, −8.741767666260650081322314167955, −7.70485280080022009321644954220, −6.96358517774413498906115097301, −6.35093168739714009787246047887, −5.30594147014309782983751405984, −3.79168104236548709721947712297, −2.26486647861347736314479366745, −0.74660356476976843643489506092,
0.61415975705748244880998787372, 1.98702355033249631736341385695, 3.97961139116920592315341015289, 4.77316301108003497696524047349, 5.84187636655994374988098432864, 6.91595351387466538515570716472, 8.103514011465418488407690183701, 8.449038253644911719154263833697, 9.517255235224670484746352679875, 10.18002106116656523025449234817
