| L(s) = 1 | + (−19.5 − 11.3i)2-s + (169. − 173. i)3-s + (255. + 443. i)4-s + (1.21e3 − 698. i)5-s + (−5.29e3 + 1.48e3i)6-s + (−1.37e4 + 2.38e4i)7-s − 1.15e4i·8-s + (−1.31e3 − 5.90e4i)9-s − 3.16e4·10-s + (2.41e5 + 1.39e5i)11-s + (1.20e5 + 3.08e4i)12-s + (−5.47e4 − 9.47e4i)13-s + (5.39e5 − 3.11e5i)14-s + (8.42e4 − 3.28e5i)15-s + (−1.31e5 + 2.27e5i)16-s + 2.53e5i·17-s + ⋯ |
| L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.699 − 0.714i)3-s + (0.249 + 0.433i)4-s + (0.387 − 0.223i)5-s + (−0.680 + 0.190i)6-s + (−0.818 + 1.41i)7-s − 0.353i·8-s + (−0.0223 − 0.999i)9-s − 0.316·10-s + (1.49 + 0.865i)11-s + (0.484 + 0.124i)12-s + (−0.147 − 0.255i)13-s + (1.00 − 0.578i)14-s + (0.110 − 0.433i)15-s + (−0.125 + 0.216i)16-s + 0.178i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.780 - 0.625i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.780 - 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.0154843 + 0.0440682i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0154843 + 0.0440682i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (19.5 + 11.3i)T \) |
| 3 | \( 1 + (-169. + 173. i)T \) |
| 5 | \( 1 + (-1.21e3 + 698. i)T \) |
| good | 7 | \( 1 + (1.37e4 - 2.38e4i)T + (-1.41e8 - 2.44e8i)T^{2} \) |
| 11 | \( 1 + (-2.41e5 - 1.39e5i)T + (1.29e10 + 2.24e10i)T^{2} \) |
| 13 | \( 1 + (5.47e4 + 9.47e4i)T + (-6.89e10 + 1.19e11i)T^{2} \) |
| 17 | \( 1 - 2.53e5iT - 2.01e12T^{2} \) |
| 19 | \( 1 + 4.61e6T + 6.13e12T^{2} \) |
| 23 | \( 1 + (1.38e6 - 8.02e5i)T + (2.07e13 - 3.58e13i)T^{2} \) |
| 29 | \( 1 + (1.52e7 + 8.78e6i)T + (2.10e14 + 3.64e14i)T^{2} \) |
| 31 | \( 1 + (1.21e7 + 2.10e7i)T + (-4.09e14 + 7.09e14i)T^{2} \) |
| 37 | \( 1 + 2.74e7T + 4.80e15T^{2} \) |
| 41 | \( 1 + (9.92e7 - 5.72e7i)T + (6.71e15 - 1.16e16i)T^{2} \) |
| 43 | \( 1 + (-8.76e7 + 1.51e8i)T + (-1.08e16 - 1.87e16i)T^{2} \) |
| 47 | \( 1 + (2.20e8 + 1.27e8i)T + (2.62e16 + 4.55e16i)T^{2} \) |
| 53 | \( 1 + 2.33e8iT - 1.74e17T^{2} \) |
| 59 | \( 1 + (-3.36e8 + 1.94e8i)T + (2.55e17 - 4.42e17i)T^{2} \) |
| 61 | \( 1 + (5.76e8 - 9.97e8i)T + (-3.56e17 - 6.17e17i)T^{2} \) |
| 67 | \( 1 + (-5.55e8 - 9.61e8i)T + (-9.11e17 + 1.57e18i)T^{2} \) |
| 71 | \( 1 + 2.28e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 + 1.52e9T + 4.29e18T^{2} \) |
| 79 | \( 1 + (2.77e9 - 4.80e9i)T + (-4.73e18 - 8.19e18i)T^{2} \) |
| 83 | \( 1 + (3.73e9 + 2.15e9i)T + (7.75e18 + 1.34e19i)T^{2} \) |
| 89 | \( 1 - 6.44e9iT - 3.11e19T^{2} \) |
| 97 | \( 1 + (3.71e9 - 6.42e9i)T + (-3.68e19 - 6.38e19i)T^{2} \) |
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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
−12.48307319190568179094578357829, −11.73971672146378199968742906276, −9.895605492426387873655722913909, −9.129659078039748197722132958290, −8.460761242226196410022993375380, −6.89511863879764488040875661735, −6.04116153954419694186017929115, −3.84304396562047495208096372042, −2.38257972854164462134093478341, −1.70440767794467180876188513723,
0.01218715531284304903098691355, 1.58658715355346887045307208913, 3.31705088068505197546302871249, 4.30716318145302191968038826131, 6.24069536830962418310643840847, 7.12584966624153052607406652950, 8.578779171637002075856562477989, 9.411011422422985529575291800638, 10.35089856168900987463307655483, 11.10843579823577547581819246731
