| L(s) = 1 | + (−1.96 − 3.41i)2-s + (18.4 + 31.9i)3-s + (56.2 − 97.4i)4-s + 280.·5-s + (72.5 − 125. i)6-s + (171.5 − 297. i)7-s − 947.·8-s + (414. − 718. i)9-s + (−553. − 957. i)10-s + (−2.74e3 − 4.75e3i)11-s + 4.14e3·12-s + (2.58e3 − 7.48e3i)13-s − 1.35e3·14-s + (5.17e3 + 8.96e3i)15-s + (−5.33e3 − 9.23e3i)16-s + (−334. + 578. i)17-s + ⋯ |
| L(s) = 1 | + (−0.174 − 0.301i)2-s + (0.393 + 0.682i)3-s + (0.439 − 0.761i)4-s + 1.00·5-s + (0.137 − 0.237i)6-s + (0.188 − 0.327i)7-s − 0.654·8-s + (0.189 − 0.328i)9-s + (−0.174 − 0.302i)10-s + (−0.621 − 1.07i)11-s + 0.692·12-s + (0.326 − 0.945i)13-s − 0.131·14-s + (0.395 + 0.685i)15-s + (−0.325 − 0.563i)16-s + (−0.0164 + 0.0285i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0638 + 0.997i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.0638 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.67718 - 1.78786i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.67718 - 1.78786i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-171.5 + 297. i)T \) |
| 13 | \( 1 + (-2.58e3 + 7.48e3i)T \) |
| good | 2 | \( 1 + (1.96 + 3.41i)T + (-64 + 110. i)T^{2} \) |
| 3 | \( 1 + (-18.4 - 31.9i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 - 280.T + 7.81e4T^{2} \) |
| 11 | \( 1 + (2.74e3 + 4.75e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 17 | \( 1 + (334. - 578. i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (2.88e4 - 5.00e4i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (5.78e3 + 1.00e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (3.69e4 + 6.39e4i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 - 1.28e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + (2.31e5 + 4.01e5i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 + (-8.72e4 - 1.51e5i)T + (-9.73e10 + 1.68e11i)T^{2} \) |
| 43 | \( 1 + (-2.79e5 + 4.83e5i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 - 4.44e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 3.75e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + (-4.72e5 + 8.19e5i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-9.75e3 + 1.68e4i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-9.20e4 - 1.59e5i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (2.61e5 - 4.52e5i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 - 4.25e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 5.64e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 4.99e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (-4.48e6 - 7.77e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + (4.02e5 - 6.97e5i)T + (-4.03e13 - 6.99e13i)T^{2} \) |
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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
−12.37996332132401356810349615650, −10.72140135793813019639686488337, −10.37102971325732797225355509990, −9.397203089881700401045091965810, −8.178848188193639003758089344484, −6.26866967331843962276094199254, −5.49716514897745767039981344433, −3.63909104667793525190077018640, −2.20196021069441349927880829884, −0.74383807135725459704226092685,
1.86555766091959259053100347267, 2.55538669091725950559394003959, 4.71879185196499292513940131360, 6.44174396314818307140847104717, 7.23170392705046365333805978006, 8.380562415456606626521914444558, 9.398515270181454309359223554172, 10.82225491107233489972579382554, 12.13274337260623139600212540261, 13.07946921763178741096043104444
