| L(s) = 1 | + (−102. + 33.2i)2-s + (−276. − 200. i)3-s + (6.03e3 − 4.38e3i)4-s + (−6.92e3 + 2.13e4i)5-s + (3.49e4 + 1.13e4i)6-s + (1.28e5 + 1.76e5i)7-s + (−2.12e5 + 2.92e5i)8-s + (−1.28e5 − 3.94e5i)9-s − 2.40e6i·10-s + (−9.64e5 + 1.48e6i)11-s − 2.54e6·12-s + (2.85e6 − 9.26e5i)13-s + (−1.89e7 − 1.37e7i)14-s + (6.19e6 − 4.50e6i)15-s + (2.55e6 − 7.87e6i)16-s + (−2.47e7 − 8.03e6i)17-s + ⋯ |
| L(s) = 1 | + (−1.59 + 0.518i)2-s + (−0.379 − 0.275i)3-s + (1.47 − 1.07i)4-s + (−0.443 + 1.36i)5-s + (0.748 + 0.243i)6-s + (1.09 + 1.50i)7-s + (−0.809 + 1.11i)8-s + (−0.241 − 0.742i)9-s − 2.40i·10-s + (−0.544 + 0.838i)11-s − 0.853·12-s + (0.590 − 0.192i)13-s + (−2.52 − 1.83i)14-s + (0.543 − 0.395i)15-s + (0.152 − 0.469i)16-s + (−1.02 − 0.332i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.763 + 0.646i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.763 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(0.0790314 - 0.215623i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0790314 - 0.215623i\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (9.64e5 - 1.48e6i)T \) |
| good | 2 | \( 1 + (102. - 33.2i)T + (3.31e3 - 2.40e3i)T^{2} \) |
| 3 | \( 1 + (276. + 200. i)T + (1.64e5 + 5.05e5i)T^{2} \) |
| 5 | \( 1 + (6.92e3 - 2.13e4i)T + (-1.97e8 - 1.43e8i)T^{2} \) |
| 7 | \( 1 + (-1.28e5 - 1.76e5i)T + (-4.27e9 + 1.31e10i)T^{2} \) |
| 13 | \( 1 + (-2.85e6 + 9.26e5i)T + (1.88e13 - 1.36e13i)T^{2} \) |
| 17 | \( 1 + (2.47e7 + 8.03e6i)T + (4.71e14 + 3.42e14i)T^{2} \) |
| 19 | \( 1 + (1.37e7 - 1.89e7i)T + (-6.83e14 - 2.10e15i)T^{2} \) |
| 23 | \( 1 + 2.03e7T + 2.19e16T^{2} \) |
| 29 | \( 1 + (2.97e8 + 4.09e8i)T + (-1.09e17 + 3.36e17i)T^{2} \) |
| 31 | \( 1 + (3.09e7 + 9.52e7i)T + (-6.37e17 + 4.62e17i)T^{2} \) |
| 37 | \( 1 + (-3.00e8 + 2.18e8i)T + (2.03e18 - 6.26e18i)T^{2} \) |
| 41 | \( 1 + (-5.62e7 + 7.74e7i)T + (-6.97e18 - 2.14e19i)T^{2} \) |
| 43 | \( 1 + 1.17e10iT - 3.99e19T^{2} \) |
| 47 | \( 1 + (-3.70e9 - 2.69e9i)T + (3.59e19 + 1.10e20i)T^{2} \) |
| 53 | \( 1 + (-1.37e8 - 4.21e8i)T + (-3.97e20 + 2.88e20i)T^{2} \) |
| 59 | \( 1 + (1.02e10 - 7.47e9i)T + (5.49e20 - 1.69e21i)T^{2} \) |
| 61 | \( 1 + (3.68e10 + 1.19e10i)T + (2.14e21 + 1.56e21i)T^{2} \) |
| 67 | \( 1 + 1.66e11T + 8.18e21T^{2} \) |
| 71 | \( 1 + (1.67e10 - 5.16e10i)T + (-1.32e22 - 9.64e21i)T^{2} \) |
| 73 | \( 1 + (9.50e10 + 1.30e11i)T + (-7.07e21 + 2.17e22i)T^{2} \) |
| 79 | \( 1 + (-2.26e11 + 7.35e10i)T + (4.78e22 - 3.47e22i)T^{2} \) |
| 83 | \( 1 + (-1.21e11 - 3.94e10i)T + (8.64e22 + 6.28e22i)T^{2} \) |
| 89 | \( 1 - 3.52e11T + 2.46e23T^{2} \) |
| 97 | \( 1 + (-6.99e10 - 2.15e11i)T + (-5.61e23 + 4.07e23i)T^{2} \) |
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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
−18.20166520547606434814651702059, −17.63547881028273928278232500203, −15.44222329296038831590729096093, −15.00100161396061209667452977672, −11.80636623737240457593727686267, −10.74886020702262643130592393644, −8.928196798765437608633942416798, −7.53127899725153250289458163876, −6.13206485728894618438998864064, −2.14325118710552251213867544847,
0.20047551267789182283446754292, 1.38959859466517459693213545331, 4.60097878648683396057564487853, 7.81790884573346086044041570273, 8.685279368562279329309614056377, 10.69087880490414425582186487585, 11.30096073285980603195282756095, 13.43404713895298219488494885398, 16.22470766523809260704746140083, 16.79924021267888357996001498789
