| L(s) = 1 | + (0.940 − 1.05i)2-s + (−0.540 − 0.841i)3-s + (−0.232 − 1.98i)4-s + (−2.80 + 1.28i)5-s + (−1.39 − 0.219i)6-s + (−0.413 + 0.121i)7-s + (−2.31 − 1.62i)8-s + (−0.415 + 0.909i)9-s + (−1.28 + 4.17i)10-s + (−2.98 + 2.58i)11-s + (−1.54 + 1.26i)12-s + (0.0814 − 0.277i)13-s + (−0.260 + 0.551i)14-s + (2.59 + 1.66i)15-s + (−3.89 + 0.924i)16-s + (−0.737 + 5.13i)17-s + ⋯ |
| L(s) = 1 | + (0.664 − 0.747i)2-s + (−0.312 − 0.485i)3-s + (−0.116 − 0.993i)4-s + (−1.25 + 0.573i)5-s + (−0.570 − 0.0896i)6-s + (−0.156 + 0.0459i)7-s + (−0.819 − 0.573i)8-s + (−0.138 + 0.303i)9-s + (−0.406 + 1.31i)10-s + (−0.899 + 0.779i)11-s + (−0.446 + 0.366i)12-s + (0.0225 − 0.0769i)13-s + (−0.0696 + 0.147i)14-s + (0.670 + 0.430i)15-s + (−0.972 + 0.231i)16-s + (−0.178 + 1.24i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0317524 + 0.0397195i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0317524 + 0.0397195i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.940 + 1.05i)T \) |
| 3 | \( 1 + (0.540 + 0.841i)T \) |
| 23 | \( 1 + (1.29 + 4.61i)T \) |
| good | 5 | \( 1 + (2.80 - 1.28i)T + (3.27 - 3.77i)T^{2} \) |
| 7 | \( 1 + (0.413 - 0.121i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (2.98 - 2.58i)T + (1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.0814 + 0.277i)T + (-10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (0.737 - 5.13i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (-2.87 + 0.413i)T + (18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (5.51 + 0.793i)T + (27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (3.14 + 2.02i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (8.93 + 4.08i)T + (24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (-0.861 - 1.88i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-2.21 - 3.44i)T + (-17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + 1.86T + 47T^{2} \) |
| 53 | \( 1 + (-0.329 - 1.12i)T + (-44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (-2.87 + 9.80i)T + (-49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (-1.41 + 2.19i)T + (-25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (3.91 + 3.39i)T + (9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (6.35 - 7.33i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (1.86 + 12.9i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (-13.5 - 3.97i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (-9.95 - 4.54i)T + (54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (7.56 - 4.86i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (3.88 + 8.51i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−11.00409700177727470199088884498, −10.66160889890552997002816516634, −9.557256038194489729769440914012, −8.193906908700752283088532763447, −7.37220054145809931261639741967, −6.43732022611332076277454721685, −5.33553985445869495671297325065, −4.23771988082411370435487498344, −3.28725190957762936439093309494, −2.01896811456670370383784651248,
0.02325990057813177547014256032, 3.15006386990757227978258457554, 3.91040275943229919608921457458, 5.05687367418603749659329870501, 5.55305981327361548571319552114, 7.02345044284461442704536038396, 7.71254786242760613503453631334, 8.575607514032681021038996854245, 9.409193433058672948165987045533, 10.79569211567963457214888247306
