| L(s) = 1 | + (1.16 + 0.866i)2-s + (1.68 − 0.404i)3-s + (0.0305 + 0.102i)4-s + (−2.88 − 1.90i)5-s + (2.31 + 0.989i)6-s + (0.453 + 2.60i)7-s + (0.939 − 2.58i)8-s + (2.67 − 1.36i)9-s + (−1.71 − 4.71i)10-s + (1.50 − 3.00i)11-s + (0.0927 + 0.159i)12-s + (6.12 + 2.64i)13-s + (−1.73 + 3.42i)14-s + (−5.63 − 2.03i)15-s + (3.51 − 2.30i)16-s + (−1.24 − 1.04i)17-s + ⋯ |
| L(s) = 1 | + (0.823 + 0.612i)2-s + (0.972 − 0.233i)3-s + (0.0152 + 0.0510i)4-s + (−1.29 − 0.849i)5-s + (0.943 + 0.403i)6-s + (0.171 + 0.985i)7-s + (0.332 − 0.913i)8-s + (0.891 − 0.453i)9-s + (−0.542 − 1.49i)10-s + (0.455 − 0.906i)11-s + (0.0267 + 0.0460i)12-s + (1.69 + 0.732i)13-s + (−0.462 + 0.916i)14-s + (−1.45 − 0.524i)15-s + (0.877 − 0.577i)16-s + (−0.302 − 0.254i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.315i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 + 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.59960 - 0.420846i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.59960 - 0.420846i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.68 + 0.404i)T \) |
| 7 | \( 1 + (-0.453 - 2.60i)T \) |
| good | 2 | \( 1 + (-1.16 - 0.866i)T + (0.573 + 1.91i)T^{2} \) |
| 5 | \( 1 + (2.88 + 1.90i)T + (1.98 + 4.59i)T^{2} \) |
| 11 | \( 1 + (-1.50 + 3.00i)T + (-6.56 - 8.82i)T^{2} \) |
| 13 | \( 1 + (-6.12 - 2.64i)T + (8.92 + 9.45i)T^{2} \) |
| 17 | \( 1 + (1.24 + 1.04i)T + (2.95 + 16.7i)T^{2} \) |
| 19 | \( 1 + (4.77 + 5.69i)T + (-3.29 + 18.7i)T^{2} \) |
| 23 | \( 1 + (0.137 + 0.129i)T + (1.33 + 22.9i)T^{2} \) |
| 29 | \( 1 + (0.561 - 4.80i)T + (-28.2 - 6.68i)T^{2} \) |
| 31 | \( 1 + (1.01 - 4.28i)T + (-27.7 - 13.9i)T^{2} \) |
| 37 | \( 1 + (1.13 - 6.41i)T + (-34.7 - 12.6i)T^{2} \) |
| 41 | \( 1 + (-3.28 - 4.41i)T + (-11.7 + 39.2i)T^{2} \) |
| 43 | \( 1 + (-0.172 - 2.96i)T + (-42.7 + 4.99i)T^{2} \) |
| 47 | \( 1 + (0.214 - 0.0507i)T + (42.0 - 21.0i)T^{2} \) |
| 53 | \( 1 + (10.0 + 5.78i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.16 + 1.58i)T + (35.2 - 47.3i)T^{2} \) |
| 61 | \( 1 + (-3.34 - 1.00i)T + (50.9 + 33.5i)T^{2} \) |
| 67 | \( 1 + (8.44 - 0.986i)T + (65.1 - 15.4i)T^{2} \) |
| 71 | \( 1 + (-4.32 - 11.8i)T + (-54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (3.67 - 10.0i)T + (-55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-1.20 + 1.62i)T + (-22.6 - 75.6i)T^{2} \) |
| 83 | \( 1 + (3.98 - 5.35i)T + (-23.8 - 79.5i)T^{2} \) |
| 89 | \( 1 + (-15.6 - 5.70i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-0.686 - 1.04i)T + (-38.4 + 89.0i)T^{2} \) |
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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
−11.01742921597995871205001754097, −9.312715103519647062094841308225, −8.630051022865765302680777669268, −8.325679097643497969685648239144, −6.92224843602013294107037090896, −6.24330644629469530550930780183, −4.88695469796050892416568213897, −4.12825180808705260347715931235, −3.24249746583827790710041010861, −1.26179159069162785756932612934,
1.96428114165444194355228028195, 3.44969885993031422295789568834, 3.87460137823059332395598972639, 4.39138812421290963516269624126, 6.27733208034363688139460931517, 7.68778880066863020979295296436, 7.80694657983995418277544513267, 8.882086122340386965268792532626, 10.49111761197079827930206908819, 10.67782346201034144488090870062
