| L(s) = 1 | + (−1.06 + 2.09i)2-s + (2.55 + 1.66i)3-s + (−2.06 − 2.84i)4-s + (1.70 − 1.45i)5-s + (−6.20 + 3.58i)6-s + (0.386 + 1.00i)7-s + (3.52 − 0.557i)8-s + (2.56 + 5.76i)9-s + (1.22 + 5.10i)10-s + (−4.03 + 0.424i)11-s + (−0.561 − 10.7i)12-s + (0.291 − 5.56i)13-s + (−2.51 − 0.264i)14-s + (6.76 − 0.887i)15-s + (−0.413 + 1.27i)16-s + (−0.995 − 1.22i)17-s + ⋯ |
| L(s) = 1 | + (−0.754 + 1.47i)2-s + (1.47 + 0.959i)3-s + (−1.03 − 1.42i)4-s + (0.760 − 0.649i)5-s + (−2.53 + 1.46i)6-s + (0.146 + 0.380i)7-s + (1.24 − 0.197i)8-s + (0.855 + 1.92i)9-s + (0.387 + 1.61i)10-s + (−1.21 + 0.127i)11-s + (−0.162 − 3.09i)12-s + (0.0809 − 1.54i)13-s + (−0.673 − 0.0707i)14-s + (1.74 − 0.229i)15-s + (−0.103 + 0.318i)16-s + (−0.241 − 0.298i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.620 - 0.784i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.620 - 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.529550 + 1.09406i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.529550 + 1.09406i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.70 + 1.45i)T \) |
| 31 | \( 1 + (5.54 - 0.517i)T \) |
| good | 2 | \( 1 + (1.06 - 2.09i)T + (-1.17 - 1.61i)T^{2} \) |
| 3 | \( 1 + (-2.55 - 1.66i)T + (1.22 + 2.74i)T^{2} \) |
| 7 | \( 1 + (-0.386 - 1.00i)T + (-5.20 + 4.68i)T^{2} \) |
| 11 | \( 1 + (4.03 - 0.424i)T + (10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (-0.291 + 5.56i)T + (-12.9 - 1.35i)T^{2} \) |
| 17 | \( 1 + (0.995 + 1.22i)T + (-3.53 + 16.6i)T^{2} \) |
| 19 | \( 1 + (2.79 + 2.51i)T + (1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (-0.305 - 1.92i)T + (-21.8 + 7.10i)T^{2} \) |
| 29 | \( 1 + (-1.18 - 3.66i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (-7.74 - 2.07i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (1.63 + 0.347i)T + (37.4 + 16.6i)T^{2} \) |
| 43 | \( 1 + (5.72 - 0.300i)T + (42.7 - 4.49i)T^{2} \) |
| 47 | \( 1 + (5.25 - 2.67i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (-10.2 - 3.93i)T + (39.3 + 35.4i)T^{2} \) |
| 59 | \( 1 + (-1.14 - 5.40i)T + (-53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 - 7.50iT - 61T^{2} \) |
| 67 | \( 1 + (-3.11 + 0.833i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (6.71 - 2.98i)T + (47.5 - 52.7i)T^{2} \) |
| 73 | \( 1 + (-1.83 + 2.26i)T + (-15.1 - 71.4i)T^{2} \) |
| 79 | \( 1 + (-0.241 + 2.30i)T + (-77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (1.99 + 3.06i)T + (-33.7 + 75.8i)T^{2} \) |
| 89 | \( 1 + (0.253 - 0.183i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (1.55 + 0.245i)T + (92.2 + 29.9i)T^{2} \) |
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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
−13.57807691963101707287896341657, −12.98868374388543627667400297991, −10.48056128207889425740760764802, −9.834809525342079341156483728914, −8.846485382906859158857180183009, −8.378699099718242430447508735174, −7.45582954883776023260251131061, −5.62995406110675936754750472283, −4.85575326990037950362155434025, −2.72257301099410810207106687682,
1.83728681334034522111690501691, 2.51599166056024699370397421800, 3.82829141797833312954971786290, 6.56390531015152039161565437369, 7.77447526445647342950140702345, 8.682179364849594087324356134093, 9.567886165479349794888475822252, 10.43356131593715060560751537872, 11.47686872785457598425276952485, 12.78389550958949737450560921472
