| L(s) = 1 | + (−0.362 + 2.52i)2-s + (0.804 + 1.76i)3-s + (−4.30 − 1.26i)4-s + (−0.654 − 0.755i)5-s + (−4.73 + 1.38i)6-s + (0.564 + 0.362i)7-s + (2.63 − 5.77i)8-s + (−0.490 + 0.565i)9-s + (2.14 − 1.37i)10-s + (0.677 + 4.70i)11-s + (−1.23 − 8.60i)12-s + (2.77 − 1.78i)13-s + (−1.11 + 1.29i)14-s + (0.804 − 1.76i)15-s + (6.04 + 3.88i)16-s + (3.87 − 1.13i)17-s + ⋯ |
| L(s) = 1 | + (−0.256 + 1.78i)2-s + (0.464 + 1.01i)3-s + (−2.15 − 0.632i)4-s + (−0.292 − 0.337i)5-s + (−1.93 + 0.567i)6-s + (0.213 + 0.137i)7-s + (0.932 − 2.04i)8-s + (−0.163 + 0.188i)9-s + (0.677 − 0.435i)10-s + (0.204 + 1.41i)11-s + (−0.357 − 2.48i)12-s + (0.769 − 0.494i)13-s + (−0.299 + 0.345i)14-s + (0.207 − 0.454i)15-s + (1.51 + 0.971i)16-s + (0.940 − 0.276i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0710i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0710i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0333424 + 0.937450i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0333424 + 0.937450i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.654 + 0.755i)T \) |
| 23 | \( 1 + (2.45 - 4.12i)T \) |
| good | 2 | \( 1 + (0.362 - 2.52i)T + (-1.91 - 0.563i)T^{2} \) |
| 3 | \( 1 + (-0.804 - 1.76i)T + (-1.96 + 2.26i)T^{2} \) |
| 7 | \( 1 + (-0.564 - 0.362i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (-0.677 - 4.70i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (-2.77 + 1.78i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-3.87 + 1.13i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (7.66 + 2.25i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (-6.35 + 1.86i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-2.31 + 5.07i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (1.02 - 1.17i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (3.37 + 3.89i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-0.646 - 1.41i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 7.40T + 47T^{2} \) |
| 53 | \( 1 + (1.43 + 0.919i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (9.89 - 6.35i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-0.248 + 0.544i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-1.56 + 10.8i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (-1.66 + 11.5i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (3.31 + 0.974i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (0.900 - 0.578i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (0.985 - 1.13i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (1.31 + 2.87i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (2.41 + 2.79i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−14.63506898440471025978889291884, −13.54181254528777009597908793791, −12.26948456592815295988598508535, −10.35046753129891406007392610909, −9.464854404863598944741905495704, −8.580496039638904955706979686291, −7.67028235666954208666128691237, −6.36066458887048695068653744616, −4.94018099302248791250807186694, −4.08044123102795534053522181942,
1.34579247010145703057183308297, 2.85633990441384309899170618810, 4.12030712284635204182077342983, 6.41858457549771526322714199258, 8.263246794439530496897782015552, 8.587675597008933229775084393253, 10.34696192488885233626244698702, 10.96867577784525460487668183094, 12.09659246046724227066622255037, 12.75638818892815098372139421426
