L(s) = 1 | + (−2.19 − 1.59i)3-s + (0.548 − 2.16i)5-s + 4.40·7-s + (1.34 + 4.15i)9-s + (0.516 − 1.58i)11-s + (−0.795 − 2.44i)13-s + (−4.66 + 3.88i)15-s + (−0.239 + 0.173i)17-s + (−3.13 + 2.28i)19-s + (−9.66 − 7.02i)21-s + (2.71 − 8.36i)23-s + (−4.39 − 2.37i)25-s + (1.14 − 3.52i)27-s + (−0.177 − 0.129i)29-s + (−5.90 + 4.29i)31-s + ⋯ |
L(s) = 1 | + (−1.26 − 0.921i)3-s + (0.245 − 0.969i)5-s + 1.66·7-s + (0.449 + 1.38i)9-s + (0.155 − 0.478i)11-s + (−0.220 − 0.679i)13-s + (−1.20 + 1.00i)15-s + (−0.0579 + 0.0421i)17-s + (−0.720 + 0.523i)19-s + (−2.10 − 1.53i)21-s + (0.566 − 1.74i)23-s + (−0.879 − 0.475i)25-s + (0.220 − 0.678i)27-s + (−0.0329 − 0.0239i)29-s + (−1.06 + 0.771i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.529 + 0.848i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.529 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.489563 - 0.882710i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.489563 - 0.882710i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.548 + 2.16i)T \) |
good | 3 | \( 1 + (2.19 + 1.59i)T + (0.927 + 2.85i)T^{2} \) |
| 7 | \( 1 - 4.40T + 7T^{2} \) |
| 11 | \( 1 + (-0.516 + 1.58i)T + (-8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (0.795 + 2.44i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (0.239 - 0.173i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (3.13 - 2.28i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-2.71 + 8.36i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (0.177 + 0.129i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (5.90 - 4.29i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (0.231 + 0.711i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (0.947 + 2.91i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 0.913T + 43T^{2} \) |
| 47 | \( 1 + (-0.00570 - 0.00414i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-9.20 - 6.68i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (0.635 + 1.95i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (3.52 - 10.8i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (-4.94 + 3.59i)T + (20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (6.51 + 4.73i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-1.08 + 3.33i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-9.42 - 6.84i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-2.30 + 1.67i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (1.02 - 3.16i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (8.73 + 6.34i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.98615250021305925745469933407, −10.55524836811569140112690460257, −8.816894514252280653518310502621, −8.181131893825859751181472983670, −7.19186062739387840177900305396, −5.97188372064384007313466676469, −5.27124311631413628922263690876, −4.46070195035396980810975051770, −1.94327408557532789611753685901, −0.820305264946501133612021259195,
1.93501076475691125092376705420, 3.91319614222806791580224939632, 4.86628235708050616583709675548, 5.61712103280120050571422584123, 6.77988226441881100123373212033, 7.67611383476608770858154101606, 9.138665165480040053626193381375, 10.00269062860857646984487502063, 10.96076668071952366826608809742, 11.31506098651296269307670260105