| L(s) = 1 | + (−28.4 + 16.4i)2-s + (286. + 495. i)3-s + (−483. + 837. i)4-s + 6.96e3i·5-s + (−1.62e4 − 9.40e3i)6-s + (6.80e4 + 3.93e4i)7-s − 9.91e4i·8-s + (−7.52e4 + 1.30e5i)9-s + (−1.14e5 − 1.98e5i)10-s + (−7.92e4 + 4.57e4i)11-s − 5.53e5·12-s + (−1.21e6 + 5.68e5i)13-s − 2.58e6·14-s + (−3.45e6 + 1.99e6i)15-s + (6.38e5 + 1.10e6i)16-s + (4.28e6 − 7.42e6i)17-s + ⋯ |
| L(s) = 1 | + (−0.629 + 0.363i)2-s + (0.679 + 1.17i)3-s + (−0.236 + 0.409i)4-s + 0.996i·5-s + (−0.855 − 0.493i)6-s + (1.53 + 0.884i)7-s − 1.06i·8-s + (−0.424 + 0.735i)9-s + (−0.361 − 0.626i)10-s + (−0.148 + 0.0856i)11-s − 0.642·12-s + (−0.905 + 0.424i)13-s − 1.28·14-s + (−1.17 + 0.677i)15-s + (0.152 + 0.263i)16-s + (0.732 − 1.26i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0975i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.0747848 + 1.53011i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0747848 + 1.53011i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (1.21e6 - 5.68e5i)T \) |
| good | 2 | \( 1 + (28.4 - 16.4i)T + (1.02e3 - 1.77e3i)T^{2} \) |
| 3 | \( 1 + (-286. - 495. i)T + (-8.85e4 + 1.53e5i)T^{2} \) |
| 5 | \( 1 - 6.96e3iT - 4.88e7T^{2} \) |
| 7 | \( 1 + (-6.80e4 - 3.93e4i)T + (9.88e8 + 1.71e9i)T^{2} \) |
| 11 | \( 1 + (7.92e4 - 4.57e4i)T + (1.42e11 - 2.47e11i)T^{2} \) |
| 17 | \( 1 + (-4.28e6 + 7.42e6i)T + (-1.71e13 - 2.96e13i)T^{2} \) |
| 19 | \( 1 + (-1.95e6 - 1.12e6i)T + (5.82e13 + 1.00e14i)T^{2} \) |
| 23 | \( 1 + (8.00e6 + 1.38e7i)T + (-4.76e14 + 8.25e14i)T^{2} \) |
| 29 | \( 1 + (-6.98e7 - 1.21e8i)T + (-6.10e15 + 1.05e16i)T^{2} \) |
| 31 | \( 1 + 2.88e8iT - 2.54e16T^{2} \) |
| 37 | \( 1 + (1.67e8 - 9.64e7i)T + (8.89e16 - 1.54e17i)T^{2} \) |
| 41 | \( 1 + (-2.12e8 + 1.22e8i)T + (2.75e17 - 4.76e17i)T^{2} \) |
| 43 | \( 1 + (6.14e8 - 1.06e9i)T + (-4.64e17 - 8.04e17i)T^{2} \) |
| 47 | \( 1 + 7.58e8iT - 2.47e18T^{2} \) |
| 53 | \( 1 + 4.25e8T + 9.26e18T^{2} \) |
| 59 | \( 1 + (-4.20e9 - 2.42e9i)T + (1.50e19 + 2.61e19i)T^{2} \) |
| 61 | \( 1 + (1.07e9 - 1.86e9i)T + (-2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (1.75e10 - 1.01e10i)T + (6.10e19 - 1.05e20i)T^{2} \) |
| 71 | \( 1 + (6.41e9 + 3.70e9i)T + (1.15e20 + 2.00e20i)T^{2} \) |
| 73 | \( 1 - 6.69e9iT - 3.13e20T^{2} \) |
| 79 | \( 1 + 7.14e8T + 7.47e20T^{2} \) |
| 83 | \( 1 + 5.10e10iT - 1.28e21T^{2} \) |
| 89 | \( 1 + (-6.19e10 + 3.57e10i)T + (1.38e21 - 2.40e21i)T^{2} \) |
| 97 | \( 1 + (-1.15e10 - 6.66e9i)T + (3.57e21 + 6.19e21i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.91541067195090139931998390689, −16.34973514681616281878669059743, −15.00655360564448318727629982460, −14.33075546098533217334049005282, −11.76872157468642693315533947722, −10.05767669442449215983226567414, −8.850474969420219675070491636901, −7.52786395126927582518453497099, −4.69404607727820653316772233751, −2.81966539900938748954133264742,
0.918115954343590014169537635924, 1.75436625559083330997643582852, 4.98975771819312431611535948755, 7.76216607988605438853770143812, 8.522371906758520630736999018169, 10.43225347070664335012260771612, 12.21714560827139584342028652563, 13.69669140429361543118586879042, 14.62301489517119594209194189577, 17.18016863982034559021245439528
