L(s) = 1 | + 25.8·2-s − 225. i·3-s + 154.·4-s − 96.4i·5-s − 5.81e3i·6-s − 1.38e3i·7-s − 9.22e3·8-s − 3.10e4·9-s − 2.49e3i·10-s − 3.38e4i·11-s − 3.48e4i·12-s + 1.38e5·13-s − 3.57e4i·14-s − 2.17e4·15-s − 3.17e5·16-s + (1.77e5 − 2.94e5i)17-s + ⋯ |
L(s) = 1 | + 1.14·2-s − 1.60i·3-s + 0.302·4-s − 0.0690i·5-s − 1.83i·6-s − 0.218i·7-s − 0.796·8-s − 1.57·9-s − 0.0787i·10-s − 0.696i·11-s − 0.485i·12-s + 1.34·13-s − 0.248i·14-s − 0.110·15-s − 1.21·16-s + (0.516 − 0.856i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.516 + 0.856i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.516 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(1.25492 - 2.22352i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.25492 - 2.22352i\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 + (-1.77e5 + 2.94e5i)T \) |
good | 2 | \( 1 - 25.8T + 512T^{2} \) |
| 3 | \( 1 + 225. iT - 1.96e4T^{2} \) |
| 5 | \( 1 + 96.4iT - 1.95e6T^{2} \) |
| 7 | \( 1 + 1.38e3iT - 4.03e7T^{2} \) |
| 11 | \( 1 + 3.38e4iT - 2.35e9T^{2} \) |
| 13 | \( 1 - 1.38e5T + 1.06e10T^{2} \) |
| 19 | \( 1 - 4.29e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 3.52e5iT - 1.80e12T^{2} \) |
| 29 | \( 1 - 5.08e5iT - 1.45e13T^{2} \) |
| 31 | \( 1 - 8.73e6iT - 2.64e13T^{2} \) |
| 37 | \( 1 + 1.62e7iT - 1.29e14T^{2} \) |
| 41 | \( 1 - 1.12e7iT - 3.27e14T^{2} \) |
| 43 | \( 1 + 2.43e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 1.11e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 3.54e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 5.03e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.53e8iT - 1.16e16T^{2} \) |
| 67 | \( 1 + 2.44e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.78e8iT - 4.58e16T^{2} \) |
| 73 | \( 1 + 1.38e7iT - 5.88e16T^{2} \) |
| 79 | \( 1 - 5.65e8iT - 1.19e17T^{2} \) |
| 83 | \( 1 + 5.68e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 2.61e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.04e9iT - 7.60e17T^{2} \) |
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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
−16.14123043729498771055201666954, −14.20548642356358078416331606197, −13.56753107029655990512679513350, −12.56966618918404039168339444633, −11.41568620185706051552028691947, −8.618835279499712785366501784884, −6.90028123876712044213115876743, −5.56091230633203782413790025509, −3.19728578534609045569333698391, −0.980109914157138175713554947025,
3.36488073646262997266880470234, 4.51061719440444289748628591740, 5.85619878146767379171472687031, 8.845550097856706796916438367264, 10.23644690991075714738693348984, 11.71908486022706062257991033832, 13.35289992754355429388512112394, 14.78731644083838175131593335006, 15.40597795491493822170280140186, 16.67086807542993625269148968307