| L(s) = 1 | + (2.07 + 2.07i)2-s + (220. + 531. i)3-s − 2.03e3i·4-s + (−4.86e3 + 2.01e3i)5-s + (−646. + 1.56e3i)6-s + (−4.65e4 − 1.92e4i)7-s + (8.49e3 − 8.49e3i)8-s + (−1.08e5 + 1.08e5i)9-s + (−1.43e4 − 5.92e3i)10-s + (1.11e5 − 2.68e5i)11-s + (1.08e6 − 4.48e5i)12-s − 5.74e5i·13-s + (−5.66e4 − 1.36e5i)14-s + (−2.14e6 − 2.14e6i)15-s − 4.14e6·16-s + (−5.27e6 − 2.54e6i)17-s + ⋯ |
| L(s) = 1 | + (0.0459 + 0.0459i)2-s + (0.522 + 1.26i)3-s − 0.995i·4-s + (−0.696 + 0.288i)5-s + (−0.0339 + 0.0819i)6-s + (−1.04 − 0.433i)7-s + (0.0916 − 0.0916i)8-s + (−0.612 + 0.612i)9-s + (−0.0452 − 0.0187i)10-s + (0.208 − 0.503i)11-s + (1.25 − 0.520i)12-s − 0.429i·13-s + (−0.0281 − 0.0679i)14-s + (−0.728 − 0.728i)15-s − 0.987·16-s + (−0.900 − 0.435i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.502 + 0.864i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.502 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.269944 - 0.469133i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.269944 - 0.469133i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (5.27e6 + 2.54e6i)T \) |
| good | 2 | \( 1 + (-2.07 - 2.07i)T + 2.04e3iT^{2} \) |
| 3 | \( 1 + (-220. - 531. i)T + (-1.25e5 + 1.25e5i)T^{2} \) |
| 5 | \( 1 + (4.86e3 - 2.01e3i)T + (3.45e7 - 3.45e7i)T^{2} \) |
| 7 | \( 1 + (4.65e4 + 1.92e4i)T + (1.39e9 + 1.39e9i)T^{2} \) |
| 11 | \( 1 + (-1.11e5 + 2.68e5i)T + (-2.01e11 - 2.01e11i)T^{2} \) |
| 13 | \( 1 + 5.74e5iT - 1.79e12T^{2} \) |
| 19 | \( 1 + (4.86e6 + 4.86e6i)T + 1.16e14iT^{2} \) |
| 23 | \( 1 + (-1.71e7 + 4.14e7i)T + (-6.73e14 - 6.73e14i)T^{2} \) |
| 29 | \( 1 + (8.88e7 - 3.68e7i)T + (8.62e15 - 8.62e15i)T^{2} \) |
| 31 | \( 1 + (-7.76e6 - 1.87e7i)T + (-1.79e16 + 1.79e16i)T^{2} \) |
| 37 | \( 1 + (-2.67e8 - 6.46e8i)T + (-1.25e17 + 1.25e17i)T^{2} \) |
| 41 | \( 1 + (9.55e8 + 3.95e8i)T + (3.89e17 + 3.89e17i)T^{2} \) |
| 43 | \( 1 + (5.11e8 - 5.11e8i)T - 9.29e17iT^{2} \) |
| 47 | \( 1 - 9.09e8iT - 2.47e18T^{2} \) |
| 53 | \( 1 + (-7.57e8 - 7.57e8i)T + 9.26e18iT^{2} \) |
| 59 | \( 1 + (-2.23e9 + 2.23e9i)T - 3.01e19iT^{2} \) |
| 61 | \( 1 + (2.91e9 + 1.20e9i)T + (3.07e19 + 3.07e19i)T^{2} \) |
| 67 | \( 1 + 1.82e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + (-4.37e9 - 1.05e10i)T + (-1.63e20 + 1.63e20i)T^{2} \) |
| 73 | \( 1 + (-2.59e10 + 1.07e10i)T + (2.21e20 - 2.21e20i)T^{2} \) |
| 79 | \( 1 + (-1.42e10 + 3.45e10i)T + (-5.28e20 - 5.28e20i)T^{2} \) |
| 83 | \( 1 + (1.27e10 + 1.27e10i)T + 1.28e21iT^{2} \) |
| 89 | \( 1 + 5.23e10iT - 2.77e21T^{2} \) |
| 97 | \( 1 + (-1.51e11 + 6.26e10i)T + (5.05e21 - 5.05e21i)T^{2} \) |
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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
−15.53125734249698683862099893683, −14.85003188303343557004545522135, −13.39426310532015423477738542006, −11.07764561102192407505853349564, −10.09233216022616436318340212346, −8.927736033133287740185281868469, −6.62162892983790146442663270359, −4.60216117805931358771566732642, −3.18801570246553022558029680695, −0.20136472884825163806101823222,
2.14807862267911542501797364877, 3.78003665685770308831880731896, 6.70934148860342017128788614749, 7.82506947027254414600313264111, 9.061558567048303940727055812003, 11.76399478814155492587642010046, 12.69214445378456333282261506870, 13.43934143808050089024344307015, 15.38547438142027034188693925769, 16.71118344044551184018827459094
