| L(s) = 1 | + (11.2 + 19.4i)2-s + (−1.14e3 − 1.97e3i)3-s + (3.84e3 − 6.65e3i)4-s + 1.79e4·5-s + (2.56e4 − 4.44e4i)6-s + (2.81e5 − 4.87e5i)7-s + 3.56e5·8-s + (−1.81e6 + 3.13e6i)9-s + (2.02e5 + 3.50e5i)10-s + (−3.49e6 − 6.05e6i)11-s − 1.75e7·12-s + (9.43e6 + 1.46e7i)13-s + 1.26e7·14-s + (−2.05e7 − 3.56e7i)15-s + (−2.74e7 − 4.76e7i)16-s + (1.89e7 − 3.28e7i)17-s + ⋯ |
| L(s) = 1 | + (0.124 + 0.214i)2-s + (−0.904 − 1.56i)3-s + (0.469 − 0.812i)4-s + 0.515·5-s + (0.224 − 0.388i)6-s + (0.904 − 1.56i)7-s + 0.480·8-s + (−1.13 + 1.96i)9-s + (0.0639 + 0.110i)10-s + (−0.594 − 1.03i)11-s − 1.69·12-s + (0.542 + 0.840i)13-s + 0.448·14-s + (−0.465 − 0.807i)15-s + (−0.409 − 0.709i)16-s + (0.190 − 0.330i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 + 0.312i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.949 + 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.274859 - 1.71630i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.274859 - 1.71630i\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (-9.43e6 - 1.46e7i)T \) |
| good | 2 | \( 1 + (-11.2 - 19.4i)T + (-4.09e3 + 7.09e3i)T^{2} \) |
| 3 | \( 1 + (1.14e3 + 1.97e3i)T + (-7.97e5 + 1.38e6i)T^{2} \) |
| 5 | \( 1 - 1.79e4T + 1.22e9T^{2} \) |
| 7 | \( 1 + (-2.81e5 + 4.87e5i)T + (-4.84e10 - 8.39e10i)T^{2} \) |
| 11 | \( 1 + (3.49e6 + 6.05e6i)T + (-1.72e13 + 2.98e13i)T^{2} \) |
| 17 | \( 1 + (-1.89e7 + 3.28e7i)T + (-4.95e15 - 8.57e15i)T^{2} \) |
| 19 | \( 1 + (3.28e7 - 5.68e7i)T + (-2.10e16 - 3.64e16i)T^{2} \) |
| 23 | \( 1 + (-4.98e8 - 8.64e8i)T + (-2.52e17 + 4.36e17i)T^{2} \) |
| 29 | \( 1 + (-1.74e9 - 3.01e9i)T + (-5.13e18 + 8.88e18i)T^{2} \) |
| 31 | \( 1 - 2.61e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + (4.31e8 + 7.47e8i)T + (-1.21e20 + 2.10e20i)T^{2} \) |
| 41 | \( 1 + (8.87e8 + 1.53e9i)T + (-4.62e20 + 8.01e20i)T^{2} \) |
| 43 | \( 1 + (3.49e8 - 6.05e8i)T + (-8.59e20 - 1.48e21i)T^{2} \) |
| 47 | \( 1 - 4.91e10T + 5.46e21T^{2} \) |
| 53 | \( 1 - 3.91e10T + 2.60e22T^{2} \) |
| 59 | \( 1 + (-1.16e11 + 2.01e11i)T + (-5.24e22 - 9.09e22i)T^{2} \) |
| 61 | \( 1 + (1.52e10 - 2.64e10i)T + (-8.09e22 - 1.40e23i)T^{2} \) |
| 67 | \( 1 + (4.37e11 + 7.57e11i)T + (-2.74e23 + 4.74e23i)T^{2} \) |
| 71 | \( 1 + (-4.80e11 + 8.31e11i)T + (-5.82e23 - 1.00e24i)T^{2} \) |
| 73 | \( 1 - 8.68e11T + 1.67e24T^{2} \) |
| 79 | \( 1 + 7.23e11T + 4.66e24T^{2} \) |
| 83 | \( 1 + 2.62e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + (-2.45e12 - 4.24e12i)T + (-1.09e25 + 1.90e25i)T^{2} \) |
| 97 | \( 1 + (3.25e12 - 5.63e12i)T + (-3.36e25 - 5.82e25i)T^{2} \) |
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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
−16.40844811025787963328772643859, −13.93414707917528867110561901265, −13.59445377231254107257827578216, −11.46067065569138444037731999173, −10.68462169457813912509789350965, −7.68666532413305515872890732407, −6.58482386355159413531027841745, −5.31492906850534322991829037342, −1.63857208500121691959273967988, −0.815259713277858722575695630838,
2.55813068519893099960530708044, 4.57589611046617611042800334523, 5.81611256863839718179955136778, 8.536950199765382629604242768439, 10.21465560741112187742167047278, 11.41944462414192002167340648407, 12.49725430626155361628114439808, 15.11240298039157704299224284036, 15.70540525307166633396230081536, 17.22383644206338917832907513043
