| L(s) = 1 | + (22.6 − 39.1i)2-s + (1.16e3 − 670. i)3-s + (−1.02e3 − 1.77e3i)4-s + (−1.42e4 − 8.21e3i)5-s − 6.06e4i·6-s − 9.26e4·8-s + (6.33e5 − 1.09e6i)9-s + (−6.43e5 + 3.71e5i)10-s + (−4.64e5 − 8.03e5i)11-s + (−2.37e6 − 1.37e6i)12-s + 8.24e6i·13-s − 2.20e7·15-s + (−2.09e6 + 3.63e6i)16-s + (−1.17e7 + 6.77e6i)17-s + (−2.86e7 − 4.96e7i)18-s + (−3.88e7 − 2.24e7i)19-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (1.59 − 0.919i)3-s + (−0.249 − 0.433i)4-s + (−0.910 − 0.525i)5-s − 1.30i·6-s − 0.353·8-s + (1.19 − 2.06i)9-s + (−0.643 + 0.371i)10-s + (−0.261 − 0.453i)11-s + (−0.796 − 0.459i)12-s + 1.70i·13-s − 1.93·15-s + (−0.125 + 0.216i)16-s + (−0.485 + 0.280i)17-s + (−0.842 − 1.45i)18-s + (−0.826 − 0.477i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(1.094679360\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.094679360\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-22.6 + 39.1i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + (-1.16e3 + 670. i)T + (2.65e5 - 4.60e5i)T^{2} \) |
| 5 | \( 1 + (1.42e4 + 8.21e3i)T + (1.22e8 + 2.11e8i)T^{2} \) |
| 11 | \( 1 + (4.64e5 + 8.03e5i)T + (-1.56e12 + 2.71e12i)T^{2} \) |
| 13 | \( 1 - 8.24e6iT - 2.32e13T^{2} \) |
| 17 | \( 1 + (1.17e7 - 6.77e6i)T + (2.91e14 - 5.04e14i)T^{2} \) |
| 19 | \( 1 + (3.88e7 + 2.24e7i)T + (1.10e15 + 1.91e15i)T^{2} \) |
| 23 | \( 1 + (-3.53e7 + 6.12e7i)T + (-1.09e16 - 1.89e16i)T^{2} \) |
| 29 | \( 1 + 5.34e8T + 3.53e17T^{2} \) |
| 31 | \( 1 + (1.14e9 - 6.62e8i)T + (3.93e17 - 6.82e17i)T^{2} \) |
| 37 | \( 1 + (-5.25e8 + 9.10e8i)T + (-3.29e18 - 5.70e18i)T^{2} \) |
| 41 | \( 1 - 1.78e9iT - 2.25e19T^{2} \) |
| 43 | \( 1 + 4.20e9T + 3.99e19T^{2} \) |
| 47 | \( 1 + (-1.50e10 - 8.68e9i)T + (5.80e19 + 1.00e20i)T^{2} \) |
| 53 | \( 1 + (-2.46e9 - 4.27e9i)T + (-2.45e20 + 4.25e20i)T^{2} \) |
| 59 | \( 1 + (7.90e9 - 4.56e9i)T + (8.89e20 - 1.54e21i)T^{2} \) |
| 61 | \( 1 + (1.85e10 + 1.07e10i)T + (1.32e21 + 2.29e21i)T^{2} \) |
| 67 | \( 1 + (3.21e10 + 5.56e10i)T + (-4.09e21 + 7.08e21i)T^{2} \) |
| 71 | \( 1 + 9.11e10T + 1.64e22T^{2} \) |
| 73 | \( 1 + (-1.80e11 + 1.04e11i)T + (1.14e22 - 1.98e22i)T^{2} \) |
| 79 | \( 1 + (-1.52e11 + 2.64e11i)T + (-2.95e22 - 5.11e22i)T^{2} \) |
| 83 | \( 1 - 2.37e11iT - 1.06e23T^{2} \) |
| 89 | \( 1 + (9.71e10 + 5.60e10i)T + (1.23e23 + 2.13e23i)T^{2} \) |
| 97 | \( 1 + 9.61e11iT - 6.93e23T^{2} \) |
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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
−11.00106806484999934710538905353, −9.166033871937496329666252574440, −8.763411182822513607903735240514, −7.62441447642209505643345549757, −6.54459061295484301324083958048, −4.41881689118982052145015457311, −3.63397327571533935544972925743, −2.37375335241119776461314097341, −1.50880269934220191065762530943, −0.15655109880737025463265443909,
2.34872168382211670248514054953, 3.40783739811675872513969083328, 4.05123344104149895408995204871, 5.34842817877529811087814702313, 7.33986253292841201147213389527, 7.893203019620401178283626977799, 8.852762529890281252461734290536, 10.03155208535102602909868530812, 11.00795594962875034786635291307, 12.73464870996144894818608714276
