| L(s) = 1 | + (−12.6 − 7.31i)2-s + (9.88 − 45.7i)3-s + (42.9 + 74.3i)4-s + (269. − 466. i)5-s + (−459. + 506. i)6-s + (626. − 656. i)7-s + 616. i·8-s + (−1.99e3 − 903. i)9-s + (−6.81e3 + 3.93e3i)10-s + (−1.64e3 + 949. i)11-s + (3.82e3 − 1.22e3i)12-s + 431. i·13-s + (−1.27e4 + 3.72e3i)14-s + (−1.86e4 − 1.69e4i)15-s + (1.00e4 − 1.73e4i)16-s + (1.04e4 + 1.80e4i)17-s + ⋯ |
| L(s) = 1 | + (−1.11 − 0.646i)2-s + (0.211 − 0.977i)3-s + (0.335 + 0.580i)4-s + (0.962 − 1.66i)5-s + (−0.868 + 0.957i)6-s + (0.690 − 0.723i)7-s + 0.425i·8-s + (−0.910 − 0.413i)9-s + (−2.15 + 1.24i)10-s + (−0.372 + 0.215i)11-s + (0.638 − 0.204i)12-s + 0.0544i·13-s + (−1.24 + 0.363i)14-s + (−1.42 − 1.29i)15-s + (0.610 − 1.05i)16-s + (0.513 + 0.889i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0548i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0548i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0294406 + 1.07338i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0294406 + 1.07338i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-9.88 + 45.7i)T \) |
| 7 | \( 1 + (-626. + 656. i)T \) |
| good | 2 | \( 1 + (12.6 + 7.31i)T + (64 + 110. i)T^{2} \) |
| 5 | \( 1 + (-269. + 466. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 11 | \( 1 + (1.64e3 - 949. i)T + (9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 - 431. iT - 6.27e7T^{2} \) |
| 17 | \( 1 + (-1.04e4 - 1.80e4i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-2.31e4 - 1.33e4i)T + (4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-2.66e4 - 1.54e4i)T + (1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 - 4.01e4iT - 1.72e10T^{2} \) |
| 31 | \( 1 + (-2.51e4 + 1.45e4i)T + (1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 + (1.43e5 - 2.48e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 - 3.09e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 4.73e4T + 2.71e11T^{2} \) |
| 47 | \( 1 + (-3.60e5 + 6.23e5i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 + (-6.79e5 + 3.92e5i)T + (5.87e11 - 1.01e12i)T^{2} \) |
| 59 | \( 1 + (4.94e5 + 8.57e5i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-1.30e6 - 7.52e5i)T + (1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (9.97e5 + 1.72e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + 6.72e5iT - 9.09e12T^{2} \) |
| 73 | \( 1 + (3.19e6 - 1.84e6i)T + (5.52e12 - 9.56e12i)T^{2} \) |
| 79 | \( 1 + (-6.26e5 + 1.08e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + 4.44e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (7.37e5 - 1.27e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + 8.19e6iT - 8.07e13T^{2} \) |
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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
−16.79079685443887353358033275064, −14.18909875023586455149525301305, −13.09380100802071608391767572039, −11.88854359854025312749053427692, −10.14725632828928471441080795109, −8.839547819293228473902147665809, −7.88094880283420881486321246466, −5.37599058914789889687976132820, −1.75039196820534942787732108928, −0.917039237444246634636372990988,
2.80643224122375163822683261179, 5.71827319159597673740023976038, 7.43688358672638057610996811051, 9.078515464534569617620703226267, 10.08831437091812966492142886360, 11.16696719979247717291465091397, 13.95870521827378757804861967330, 14.95223877769231933679320022609, 15.93595408281406354349225487646, 17.42571931184374238675404898163
