| L(s) = 1 | + (54.9 − 95.2i)3-s + (1.11e3 − 643. i)5-s + (−1.14e3 + 6.24e3i)7-s + (3.79e3 + 6.57e3i)9-s + (−2.05e4 − 1.18e4i)11-s − 1.81e5i·13-s − 1.41e5i·15-s + (3.06e5 + 1.77e5i)17-s + (4.86e5 + 8.41e5i)19-s + (5.32e5 + 4.52e5i)21-s + (5.19e5 − 2.99e5i)23-s + (−1.47e5 + 2.55e5i)25-s + 2.99e6·27-s − 3.25e6·29-s + (2.62e6 − 4.54e6i)31-s + ⋯ |
| L(s) = 1 | + (0.391 − 0.678i)3-s + (0.797 − 0.460i)5-s + (−0.179 + 0.983i)7-s + (0.192 + 0.334i)9-s + (−0.422 − 0.244i)11-s − 1.76i·13-s − 0.722i·15-s + (0.891 + 0.514i)17-s + (0.855 + 1.48i)19-s + (0.597 + 0.507i)21-s + (0.386 − 0.223i)23-s + (−0.0754 + 0.130i)25-s + 1.08·27-s − 0.854·29-s + (0.510 − 0.884i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.837 + 0.545i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.837 + 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(3.033721707\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.033721707\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (1.14e3 - 6.24e3i)T \) |
| good | 3 | \( 1 + (-54.9 + 95.2i)T + (-9.84e3 - 1.70e4i)T^{2} \) |
| 5 | \( 1 + (-1.11e3 + 643. i)T + (9.76e5 - 1.69e6i)T^{2} \) |
| 11 | \( 1 + (2.05e4 + 1.18e4i)T + (1.17e9 + 2.04e9i)T^{2} \) |
| 13 | \( 1 + 1.81e5iT - 1.06e10T^{2} \) |
| 17 | \( 1 + (-3.06e5 - 1.77e5i)T + (5.92e10 + 1.02e11i)T^{2} \) |
| 19 | \( 1 + (-4.86e5 - 8.41e5i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (-5.19e5 + 2.99e5i)T + (9.00e11 - 1.55e12i)T^{2} \) |
| 29 | \( 1 + 3.25e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + (-2.62e6 + 4.54e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + (-8.70e6 - 1.50e7i)T + (-6.49e13 + 1.12e14i)T^{2} \) |
| 41 | \( 1 + 2.87e7iT - 3.27e14T^{2} \) |
| 43 | \( 1 + 1.33e7iT - 5.02e14T^{2} \) |
| 47 | \( 1 + (-3.28e7 - 5.68e7i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 + (-4.01e7 + 6.95e7i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (2.49e7 - 4.32e7i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (-7.48e7 + 4.31e7i)T + (5.84e15 - 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-8.35e7 - 4.82e7i)T + (1.36e16 + 2.35e16i)T^{2} \) |
| 71 | \( 1 + 9.10e7iT - 4.58e16T^{2} \) |
| 73 | \( 1 + (6.36e7 + 3.67e7i)T + (2.94e16 + 5.09e16i)T^{2} \) |
| 79 | \( 1 + (-3.83e7 + 2.21e7i)T + (5.99e16 - 1.03e17i)T^{2} \) |
| 83 | \( 1 - 7.60e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (-4.91e8 + 2.83e8i)T + (1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 + 9.77e8iT - 7.60e17T^{2} \) |
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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
−12.14204254253059944785764722575, −10.48856800980124734807282041934, −9.629208014525075685061061858066, −8.295303330142043577198094040972, −7.68668980896054414610866652891, −5.84960663338882163894666825607, −5.39287206204467895378161091552, −3.21109223722569022287366772703, −2.09184310275366406453467261249, −0.955413827741485530136873373013,
0.997314427111634526906633149130, 2.57668696367926536533717375509, 3.82121204316785945969242101138, 4.93516515863722015496237361116, 6.57970482883989008382085687082, 7.36740255880001771910195125688, 9.226009968487379341444315975015, 9.666713496957112568383283065459, 10.66173031856571931428344778251, 11.78387986275935782377499549819
