L(s) = 1 | + (1.35 + 0.784i)3-s + (−1.10 − 1.92i)5-s + (2.33 − 1.24i)7-s + (−0.269 − 0.466i)9-s + (3.30 + 0.220i)11-s − 0.689i·13-s − 3.48i·15-s + (−4.42 − 2.55i)17-s + (−3.83 − 6.64i)19-s + (4.14 + 0.137i)21-s + (−4.29 + 2.48i)23-s + (0.0360 − 0.0624i)25-s − 5.55i·27-s + 2.89i·29-s + (5.75 + 3.32i)31-s + ⋯ |
L(s) = 1 | + (0.784 + 0.452i)3-s + (−0.496 − 0.859i)5-s + (0.882 − 0.471i)7-s + (−0.0897 − 0.155i)9-s + (0.997 + 0.0664i)11-s − 0.191i·13-s − 0.899i·15-s + (−1.07 − 0.619i)17-s + (−0.880 − 1.52i)19-s + (0.905 + 0.0299i)21-s + (−0.896 + 0.517i)23-s + (0.00720 − 0.0124i)25-s − 1.06i·27-s + 0.536i·29-s + (1.03 + 0.596i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.352 + 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.352 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.965122702\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.965122702\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-2.33 + 1.24i)T \) |
| 11 | \( 1 + (-3.30 - 0.220i)T \) |
good | 3 | \( 1 + (-1.35 - 0.784i)T + (1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (1.10 + 1.92i)T + (-2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 + 0.689iT - 13T^{2} \) |
| 17 | \( 1 + (4.42 + 2.55i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.83 + 6.64i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4.29 - 2.48i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 2.89iT - 29T^{2} \) |
| 31 | \( 1 + (-5.75 - 3.32i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.36 - 7.56i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 6.78iT - 41T^{2} \) |
| 43 | \( 1 + 6.15T + 43T^{2} \) |
| 47 | \( 1 + (-4.05 + 2.33i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.52 - 2.64i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1.13 + 0.658i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.68 + 1.55i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.14 - 0.663i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 3.14iT - 71T^{2} \) |
| 73 | \( 1 + (-12.3 - 7.13i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (6.66 + 11.5i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 6.98T + 83T^{2} \) |
| 89 | \( 1 + (-2.10 - 3.64i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 10.7T + 97T^{2} \) |
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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
−9.304762600289813623511800866816, −8.641814213457991126478441149273, −8.338318201342176747863365974553, −7.18532876582837014965198996741, −6.35785808769824393307056224514, −4.78677307111276579974428127689, −4.48946662889106651754092984360, −3.53829667561545094264189384447, −2.24001979902149813301107156590, −0.76816755093466679058251823189,
1.77283455374265874494456994579, 2.46012489028587054400624454245, 3.74692524791492089801398470031, 4.45143064383885995542633442823, 5.95773728338251271870388195206, 6.58328814059757799765303756359, 7.65847206249925677123499489828, 8.219571438560374968906539905288, 8.739978760993893331142251963213, 9.817525872312074308685799342743