| L(s) = 1 | + (75.1 − 24.4i)2-s + (1.00e3 + 731. i)3-s + (1.73e3 − 1.25e3i)4-s + (7.62e3 − 2.34e4i)5-s + (9.34e4 + 3.03e4i)6-s + (1.19e4 + 1.64e4i)7-s + (−9.06e4 + 1.24e5i)8-s + (3.14e5 + 9.67e5i)9-s − 1.94e6i·10-s + (1.70e6 − 4.77e5i)11-s + 2.66e6·12-s + (−5.65e6 + 1.83e6i)13-s + (1.29e6 + 9.43e5i)14-s + (2.48e7 − 1.80e7i)15-s + (−6.47e6 + 1.99e7i)16-s + (−1.72e7 − 5.61e6i)17-s + ⋯ |
| L(s) = 1 | + (1.17 − 0.381i)2-s + (1.38 + 1.00i)3-s + (0.423 − 0.307i)4-s + (0.487 − 1.50i)5-s + (2.00 + 0.651i)6-s + (0.101 + 0.139i)7-s + (−0.345 + 0.476i)8-s + (0.591 + 1.82i)9-s − 1.94i·10-s + (0.962 − 0.269i)11-s + 0.892·12-s + (−1.17 + 0.380i)13-s + (0.172 + 0.125i)14-s + (2.18 − 1.58i)15-s + (−0.386 + 1.18i)16-s + (−0.716 − 0.232i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0190i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.999 + 0.0190i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(4.61014 - 0.0438824i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.61014 - 0.0438824i\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-1.70e6 + 4.77e5i)T \) |
| good | 2 | \( 1 + (-75.1 + 24.4i)T + (3.31e3 - 2.40e3i)T^{2} \) |
| 3 | \( 1 + (-1.00e3 - 731. i)T + (1.64e5 + 5.05e5i)T^{2} \) |
| 5 | \( 1 + (-7.62e3 + 2.34e4i)T + (-1.97e8 - 1.43e8i)T^{2} \) |
| 7 | \( 1 + (-1.19e4 - 1.64e4i)T + (-4.27e9 + 1.31e10i)T^{2} \) |
| 13 | \( 1 + (5.65e6 - 1.83e6i)T + (1.88e13 - 1.36e13i)T^{2} \) |
| 17 | \( 1 + (1.72e7 + 5.61e6i)T + (4.71e14 + 3.42e14i)T^{2} \) |
| 19 | \( 1 + (-2.82e7 + 3.89e7i)T + (-6.83e14 - 2.10e15i)T^{2} \) |
| 23 | \( 1 + 1.80e8T + 2.19e16T^{2} \) |
| 29 | \( 1 + (1.48e8 + 2.03e8i)T + (-1.09e17 + 3.36e17i)T^{2} \) |
| 31 | \( 1 + (-1.26e8 - 3.90e8i)T + (-6.37e17 + 4.62e17i)T^{2} \) |
| 37 | \( 1 + (2.48e8 - 1.80e8i)T + (2.03e18 - 6.26e18i)T^{2} \) |
| 41 | \( 1 + (2.42e9 - 3.33e9i)T + (-6.97e18 - 2.14e19i)T^{2} \) |
| 43 | \( 1 + 8.17e9iT - 3.99e19T^{2} \) |
| 47 | \( 1 + (-8.27e9 - 6.01e9i)T + (3.59e19 + 1.10e20i)T^{2} \) |
| 53 | \( 1 + (-6.46e9 - 1.99e10i)T + (-3.97e20 + 2.88e20i)T^{2} \) |
| 59 | \( 1 + (7.55e9 - 5.49e9i)T + (5.49e20 - 1.69e21i)T^{2} \) |
| 61 | \( 1 + (1.62e10 + 5.28e9i)T + (2.14e21 + 1.56e21i)T^{2} \) |
| 67 | \( 1 - 1.59e11T + 8.18e21T^{2} \) |
| 71 | \( 1 + (2.74e10 - 8.44e10i)T + (-1.32e22 - 9.64e21i)T^{2} \) |
| 73 | \( 1 + (-1.13e11 - 1.56e11i)T + (-7.07e21 + 2.17e22i)T^{2} \) |
| 79 | \( 1 + (-4.18e10 + 1.35e10i)T + (4.78e22 - 3.47e22i)T^{2} \) |
| 83 | \( 1 + (1.04e11 + 3.40e10i)T + (8.64e22 + 6.28e22i)T^{2} \) |
| 89 | \( 1 - 9.76e11T + 2.46e23T^{2} \) |
| 97 | \( 1 + (1.25e11 + 3.85e11i)T + (-5.61e23 + 4.07e23i)T^{2} \) |
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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
−17.11247555416833209549532036514, −15.63201739957738998304379451154, −14.27023924939319877611076858341, −13.49152364524463156631642652677, −12.01101118169318558465165816288, −9.500437712825371137134887063093, −8.647693276681778010476031724705, −5.03514696888111635011050111749, −4.05701013961640713585866262039, −2.26188406817707841942114343774,
2.22678382324535833109959141438, 3.58181312444173674955275436101, 6.39017991273351132470810143187, 7.43667219348351747674038344226, 9.700722896488512755803267996094, 12.27419734433217010368648952804, 13.72807484970777758801684779705, 14.39679565988521244537310632297, 15.04751958215598474718702126412, 17.87617868413914948135778683352
