| L(s) = 1 | + (0.0241 − 0.0278i)2-s + (13.1 − 8.42i)3-s + (9.10 + 63.3i)4-s + (−89.8 + 41.0i)5-s + (0.0817 − 0.568i)6-s + (−160. − 547. i)7-s + (3.96 + 2.55i)8-s + (100. − 221. i)9-s + (−1.02 + 3.49i)10-s + (1.31e3 − 1.13e3i)11-s + (653. + 753. i)12-s + (1.52e3 + 447. i)13-s + (−19.1 − 8.73i)14-s + (−832. + 1.29e3i)15-s + (−3.92e3 + 1.15e3i)16-s + (7.18e3 + 1.03e3i)17-s + ⋯ |
| L(s) = 1 | + (0.00301 − 0.00348i)2-s + (0.485 − 0.312i)3-s + (0.142 + 0.989i)4-s + (−0.718 + 0.328i)5-s + (0.000378 − 0.00263i)6-s + (−0.468 − 1.59i)7-s + (0.00775 + 0.00498i)8-s + (0.138 − 0.303i)9-s + (−0.00102 + 0.00349i)10-s + (0.987 − 0.855i)11-s + (0.378 + 0.436i)12-s + (0.693 + 0.203i)13-s + (−0.00696 − 0.00318i)14-s + (−0.246 + 0.383i)15-s + (−0.959 + 0.281i)16-s + (1.46 + 0.210i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.754 + 0.656i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.754 + 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.85006 - 0.692831i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.85006 - 0.692831i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-13.1 + 8.42i)T \) |
| 23 | \( 1 + (8.17e3 + 9.01e3i)T \) |
| good | 2 | \( 1 + (-0.0241 + 0.0278i)T + (-9.10 - 63.3i)T^{2} \) |
| 5 | \( 1 + (89.8 - 41.0i)T + (1.02e4 - 1.18e4i)T^{2} \) |
| 7 | \( 1 + (160. + 547. i)T + (-9.89e4 + 6.36e4i)T^{2} \) |
| 11 | \( 1 + (-1.31e3 + 1.13e3i)T + (2.52e5 - 1.75e6i)T^{2} \) |
| 13 | \( 1 + (-1.52e3 - 447. i)T + (4.06e6 + 2.60e6i)T^{2} \) |
| 17 | \( 1 + (-7.18e3 - 1.03e3i)T + (2.31e7 + 6.80e6i)T^{2} \) |
| 19 | \( 1 + (-1.00e4 + 1.45e3i)T + (4.51e7 - 1.32e7i)T^{2} \) |
| 29 | \( 1 + (-4.73e3 + 3.28e4i)T + (-5.70e8 - 1.67e8i)T^{2} \) |
| 31 | \( 1 + (-3.44e4 - 2.21e4i)T + (3.68e8 + 8.07e8i)T^{2} \) |
| 37 | \( 1 + (1.89e4 + 8.67e3i)T + (1.68e9 + 1.93e9i)T^{2} \) |
| 41 | \( 1 + (2.11e4 + 4.62e4i)T + (-3.11e9 + 3.58e9i)T^{2} \) |
| 43 | \( 1 + (6.63e4 + 1.03e5i)T + (-2.62e9 + 5.75e9i)T^{2} \) |
| 47 | \( 1 + 1.14e4T + 1.07e10T^{2} \) |
| 53 | \( 1 + (3.79e4 + 1.29e5i)T + (-1.86e10 + 1.19e10i)T^{2} \) |
| 59 | \( 1 + (1.23e5 + 3.63e4i)T + (3.54e10 + 2.28e10i)T^{2} \) |
| 61 | \( 1 + (1.03e5 - 1.60e5i)T + (-2.14e10 - 4.68e10i)T^{2} \) |
| 67 | \( 1 + (-3.26e5 - 2.82e5i)T + (1.28e10 + 8.95e10i)T^{2} \) |
| 71 | \( 1 + (-3.23e5 + 3.72e5i)T + (-1.82e10 - 1.26e11i)T^{2} \) |
| 73 | \( 1 + (-7.09e4 - 4.93e5i)T + (-1.45e11 + 4.26e10i)T^{2} \) |
| 79 | \( 1 + (1.45e5 - 4.95e5i)T + (-2.04e11 - 1.31e11i)T^{2} \) |
| 83 | \( 1 + (-7.93e5 - 3.62e5i)T + (2.14e11 + 2.47e11i)T^{2} \) |
| 89 | \( 1 + (3.06e5 + 4.76e5i)T + (-2.06e11 + 4.52e11i)T^{2} \) |
| 97 | \( 1 + (-8.98e5 + 4.10e5i)T + (5.45e11 - 6.29e11i)T^{2} \) |
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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
−13.71447384842965589507910880595, −12.20017034616362874831405264312, −11.39449334753789184902288469428, −9.977073345272025113350822390566, −8.392849706032504401512321600448, −7.52702958124316632128546861582, −6.60417627498976402596399358577, −3.80613315592944402080006146398, −3.42632938223636750624103710221, −0.852196263402612359538872674760,
1.43154568781810949786272852462, 3.26480222738388211967693789381, 5.03169815913631706109423273778, 6.20945288200502584971329239856, 7.944992592655995548273621192299, 9.297851453449823513512221959508, 9.859156126731680787055898397825, 11.66200074103274856560698260809, 12.24125115035090554850669914182, 13.92772877017324594188708468038
