| L(s) = 1 | + (−0.763 + 0.763i)2-s + (−66.0 + 159. i)3-s + 510. i·4-s + (−1.92e3 − 797. i)5-s + (−71.2 − 171. i)6-s + (7.14e3 − 2.96e3i)7-s + (−780. − 780. i)8-s + (−7.12e3 − 7.12e3i)9-s + (2.07e3 − 861. i)10-s + (−3.32e4 − 8.02e4i)11-s + (−8.14e4 − 3.37e4i)12-s + 1.21e5i·13-s + (−3.19e3 + 7.71e3i)14-s + (2.54e5 − 2.54e5i)15-s − 2.60e5·16-s + (−2.77e5 − 2.03e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.0337 + 0.0337i)2-s + (−0.470 + 1.13i)3-s + 0.997i·4-s + (−1.37 − 0.570i)5-s + (−0.0224 − 0.0541i)6-s + (1.12 − 0.466i)7-s + (−0.0673 − 0.0673i)8-s + (−0.361 − 0.361i)9-s + (0.0657 − 0.0272i)10-s + (−0.684 − 1.65i)11-s + (−1.13 − 0.469i)12-s + 1.17i·13-s + (−0.0222 + 0.0536i)14-s + (1.29 − 1.29i)15-s − 0.993·16-s + (−0.807 − 0.590i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.527 + 0.849i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.527 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.0347946 - 0.0625575i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0347946 - 0.0625575i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (2.77e5 + 2.03e5i)T \) |
| good | 2 | \( 1 + (0.763 - 0.763i)T - 512iT^{2} \) |
| 3 | \( 1 + (66.0 - 159. i)T + (-1.39e4 - 1.39e4i)T^{2} \) |
| 5 | \( 1 + (1.92e3 + 797. i)T + (1.38e6 + 1.38e6i)T^{2} \) |
| 7 | \( 1 + (-7.14e3 + 2.96e3i)T + (2.85e7 - 2.85e7i)T^{2} \) |
| 11 | \( 1 + (3.32e4 + 8.02e4i)T + (-1.66e9 + 1.66e9i)T^{2} \) |
| 13 | \( 1 - 1.21e5iT - 1.06e10T^{2} \) |
| 19 | \( 1 + (6.65e4 - 6.65e4i)T - 3.22e11iT^{2} \) |
| 23 | \( 1 + (-2.83e5 - 6.83e5i)T + (-1.27e12 + 1.27e12i)T^{2} \) |
| 29 | \( 1 + (3.63e6 + 1.50e6i)T + (1.02e13 + 1.02e13i)T^{2} \) |
| 31 | \( 1 + (-1.54e6 + 3.73e6i)T + (-1.86e13 - 1.86e13i)T^{2} \) |
| 37 | \( 1 + (8.05e5 - 1.94e6i)T + (-9.18e13 - 9.18e13i)T^{2} \) |
| 41 | \( 1 + (1.08e7 - 4.50e6i)T + (2.31e14 - 2.31e14i)T^{2} \) |
| 43 | \( 1 + (2.84e7 + 2.84e7i)T + 5.02e14iT^{2} \) |
| 47 | \( 1 - 4.45e7iT - 1.11e15T^{2} \) |
| 53 | \( 1 + (2.84e7 - 2.84e7i)T - 3.29e15iT^{2} \) |
| 59 | \( 1 + (-2.86e7 - 2.86e7i)T + 8.66e15iT^{2} \) |
| 61 | \( 1 + (-1.51e7 + 6.26e6i)T + (8.26e15 - 8.26e15i)T^{2} \) |
| 67 | \( 1 + 1.48e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + (8.18e7 - 1.97e8i)T + (-3.24e16 - 3.24e16i)T^{2} \) |
| 73 | \( 1 + (-2.44e8 - 1.01e8i)T + (4.16e16 + 4.16e16i)T^{2} \) |
| 79 | \( 1 + (-1.19e7 - 2.88e7i)T + (-8.47e16 + 8.47e16i)T^{2} \) |
| 83 | \( 1 + (-2.43e8 + 2.43e8i)T - 1.86e17iT^{2} \) |
| 89 | \( 1 - 3.04e7iT - 3.50e17T^{2} \) |
| 97 | \( 1 + (4.54e8 + 1.88e8i)T + (5.37e17 + 5.37e17i)T^{2} \) |
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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
−17.02255092681478529508246582137, −16.33054529767128561410247652674, −15.50493513347569771533776476820, −13.54984656357622129718180846004, −11.52063514816760868352754662236, −11.22875705170995067406844919942, −8.784698666006561158361348238281, −7.69979434814085765045278386684, −4.75070210525537558703075731963, −3.80817862030706094081512347545,
0.03756933207487716789402217147, 1.86928294246724870100065896724, 4.96314017680507510422272652968, 6.87966931144607267023285079525, 8.029763020138733673206987513759, 10.57267091399713723349910705588, 11.67105501827235603358621055812, 12.85133480764584202491117320249, 15.01487904269390880651147104526, 15.22454114709769710166014054249
