| L(s) = 1 | + (−0.654 + 0.755i)3-s + (0.457 − 3.18i)5-s + (0.0191 + 0.0419i)7-s + (−0.142 − 0.989i)9-s + (−5.15 + 1.51i)11-s + (2.40 − 5.25i)13-s + (2.10 + 2.43i)15-s + (−5.34 + 3.43i)17-s + (−3.77 − 2.42i)19-s + (−0.0442 − 0.0129i)21-s + (3.12 − 3.63i)23-s + (−5.12 − 1.50i)25-s + (0.841 + 0.540i)27-s + (−2.28 + 1.46i)29-s + (−0.106 − 0.123i)31-s + ⋯ |
| L(s) = 1 | + (−0.378 + 0.436i)3-s + (0.204 − 1.42i)5-s + (0.00723 + 0.0158i)7-s + (−0.0474 − 0.329i)9-s + (−1.55 + 0.456i)11-s + (0.666 − 1.45i)13-s + (0.543 + 0.627i)15-s + (−1.29 + 0.833i)17-s + (−0.866 − 0.556i)19-s + (−0.00964 − 0.00283i)21-s + (0.651 − 0.758i)23-s + (−1.02 − 0.301i)25-s + (0.161 + 0.104i)27-s + (−0.424 + 0.272i)29-s + (−0.0191 − 0.0221i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.517 + 0.855i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.517 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.378614 - 0.671092i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.378614 - 0.671092i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.654 - 0.755i)T \) |
| 23 | \( 1 + (-3.12 + 3.63i)T \) |
| good | 5 | \( 1 + (-0.457 + 3.18i)T + (-4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (-0.0191 - 0.0419i)T + (-4.58 + 5.29i)T^{2} \) |
| 11 | \( 1 + (5.15 - 1.51i)T + (9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (-2.40 + 5.25i)T + (-8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (5.34 - 3.43i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (3.77 + 2.42i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (2.28 - 1.46i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (0.106 + 0.123i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (0.848 + 5.89i)T + (-35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (-0.239 + 1.66i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (-1.74 + 2.00i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + 6.04T + 47T^{2} \) |
| 53 | \( 1 + (4.43 + 9.71i)T + (-34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (-4.97 + 10.8i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-7.50 - 8.66i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (-4.80 - 1.41i)T + (56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (-4.54 - 1.33i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (-3.79 - 2.44i)T + (30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (-1.83 + 4.01i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (-2.40 - 16.7i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (10.9 - 12.6i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (1.17 - 8.16i)T + (-93.0 - 27.3i)T^{2} \) |
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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
−10.64985628388363730393638870916, −9.642069102709693238344822647656, −8.538167701404897603091195156696, −8.246840031190331722875900295571, −6.74304769408220234386991326494, −5.46103359179051054297768773486, −5.06149283201906876166802615503, −3.97172865697857489865837080569, −2.30202438701005021842356494207, −0.43656938672883093058255844287,
2.06729816306672564581301142046, 3.05073371638017757579572829110, 4.51196332362118530152227628872, 5.81709974762155503747243307838, 6.61541887456962487274169534684, 7.25401255704153088338820601344, 8.298778172094829399010002718258, 9.409196758170675738662669594541, 10.50885543637432082827491831828, 11.10014741377132285692443178126
