| L(s) = 1 | + (−1.13 − 2.21i)3-s + (−0.139 − 2.23i)5-s + (0.125 + 0.125i)7-s + (−1.88 + 2.59i)9-s + (−3.73 − 5.14i)11-s + (0.413 + 2.61i)13-s + (−4.79 + 2.83i)15-s + (3.67 + 1.87i)17-s + (0.529 + 1.62i)19-s + (0.136 − 0.421i)21-s + (−0.203 + 1.28i)23-s + (−4.96 + 0.621i)25-s + (0.501 + 0.0794i)27-s + (−8.12 − 2.64i)29-s + (−1.69 + 0.551i)31-s + ⋯ |
| L(s) = 1 | + (−0.652 − 1.28i)3-s + (−0.0622 − 0.998i)5-s + (0.0475 + 0.0475i)7-s + (−0.627 + 0.864i)9-s + (−1.12 − 1.55i)11-s + (0.114 + 0.723i)13-s + (−1.23 + 0.731i)15-s + (0.891 + 0.454i)17-s + (0.121 + 0.373i)19-s + (0.0298 − 0.0919i)21-s + (−0.0425 + 0.268i)23-s + (−0.992 + 0.124i)25-s + (0.0965 + 0.0152i)27-s + (−1.50 − 0.490i)29-s + (−0.304 + 0.0989i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0407i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0407i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0155810 - 0.764176i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0155810 - 0.764176i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.139 + 2.23i)T \) |
| good | 3 | \( 1 + (1.13 + 2.21i)T + (-1.76 + 2.42i)T^{2} \) |
| 7 | \( 1 + (-0.125 - 0.125i)T + 7iT^{2} \) |
| 11 | \( 1 + (3.73 + 5.14i)T + (-3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.413 - 2.61i)T + (-12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (-3.67 - 1.87i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (-0.529 - 1.62i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (0.203 - 1.28i)T + (-21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (8.12 + 2.64i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (1.69 - 0.551i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-6.42 + 1.01i)T + (35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (8.51 + 6.18i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (2.70 - 2.70i)T - 43iT^{2} \) |
| 47 | \( 1 + (-9.15 + 4.66i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (-8.64 + 4.40i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (-4.80 - 3.49i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-6.89 + 5.00i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (1.50 - 2.95i)T + (-39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (-7.64 - 2.48i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (1.62 + 0.256i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (-2.78 + 8.57i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (9.56 + 4.87i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (-0.221 - 0.305i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (4.08 + 8.01i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−11.20877045571542016259416055071, −9.967017347242605421868701287184, −8.665289458417983567893004196579, −8.031857174834669064810482183741, −7.12393599218756099524114936067, −5.69824100608406476474420398830, −5.55785414978904493272553225286, −3.73837150197799735250671211037, −1.88988157401116459942165163924, −0.53331010677072681084459336279,
2.61475488520945180465897047853, 3.84782924792728427247225008693, 4.98500239682111795769269391929, 5.69456608906967163722956923989, 7.11729490587356872754409522357, 7.83627275181312660932012144417, 9.493831495472087853256929252468, 10.06877426700796761304835890762, 10.66057195781794015586269792775, 11.39036962468811140339694430154
