| L(s) = 1 | + (−12.2 − 10.2i)2-s + (−70.5 + 25.6i)3-s + (44.4 + 252. i)4-s + (407. − 2.31e3i)5-s + (1.12e3 + 410. i)6-s + (4.39e3 + 7.61e3i)7-s + (2.04e3 − 3.54e3i)8-s + (−1.07e4 + 9.02e3i)9-s + (−2.87e4 + 2.41e4i)10-s + (−3.97e3 + 6.88e3i)11-s + (−9.61e3 − 1.66e4i)12-s + (2.56e4 + 9.35e3i)13-s + (2.44e4 − 1.38e5i)14-s + (3.05e4 + 1.73e5i)15-s + (−6.15e4 + 2.24e4i)16-s + (2.43e5 + 2.04e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 − 0.454i)2-s + (−0.503 + 0.183i)3-s + (0.0868 + 0.492i)4-s + (0.291 − 1.65i)5-s + (0.355 + 0.129i)6-s + (0.692 + 1.19i)7-s + (0.176 − 0.306i)8-s + (−0.546 + 0.458i)9-s + (−0.909 + 0.762i)10-s + (−0.0818 + 0.141i)11-s + (−0.133 − 0.231i)12-s + (0.249 + 0.0908i)13-s + (0.169 − 0.963i)14-s + (0.156 + 0.884i)15-s + (−0.234 + 0.0855i)16-s + (0.708 + 0.594i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.314 + 0.949i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.314 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.00125 - 0.722982i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.00125 - 0.722982i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (12.2 + 10.2i)T \) |
| 19 | \( 1 + (-5.04e5 + 2.62e5i)T \) |
| good | 3 | \( 1 + (70.5 - 25.6i)T + (1.50e4 - 1.26e4i)T^{2} \) |
| 5 | \( 1 + (-407. + 2.31e3i)T + (-1.83e6 - 6.68e5i)T^{2} \) |
| 7 | \( 1 + (-4.39e3 - 7.61e3i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + (3.97e3 - 6.88e3i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 + (-2.56e4 - 9.35e3i)T + (8.12e9 + 6.81e9i)T^{2} \) |
| 17 | \( 1 + (-2.43e5 - 2.04e5i)T + (2.05e10 + 1.16e11i)T^{2} \) |
| 23 | \( 1 + (4.06e5 + 2.30e6i)T + (-1.69e12 + 6.16e11i)T^{2} \) |
| 29 | \( 1 + (-4.10e6 + 3.44e6i)T + (2.51e12 - 1.42e13i)T^{2} \) |
| 31 | \( 1 + (-1.36e6 - 2.37e6i)T + (-1.32e13 + 2.28e13i)T^{2} \) |
| 37 | \( 1 + 9.07e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + (3.84e6 - 1.39e6i)T + (2.50e14 - 2.10e14i)T^{2} \) |
| 43 | \( 1 + (-4.37e6 + 2.47e7i)T + (-4.72e14 - 1.71e14i)T^{2} \) |
| 47 | \( 1 + (-4.34e7 + 3.64e7i)T + (1.94e14 - 1.10e15i)T^{2} \) |
| 53 | \( 1 + (-3.10e6 - 1.75e7i)T + (-3.10e15 + 1.12e15i)T^{2} \) |
| 59 | \( 1 + (-3.87e7 - 3.25e7i)T + (1.50e15 + 8.53e15i)T^{2} \) |
| 61 | \( 1 + (-8.43e6 - 4.78e7i)T + (-1.09e16 + 3.99e15i)T^{2} \) |
| 67 | \( 1 + (8.11e7 - 6.80e7i)T + (4.72e15 - 2.67e16i)T^{2} \) |
| 71 | \( 1 + (-4.65e7 + 2.63e8i)T + (-4.30e16 - 1.56e16i)T^{2} \) |
| 73 | \( 1 + (-3.98e8 + 1.45e8i)T + (4.50e16 - 3.78e16i)T^{2} \) |
| 79 | \( 1 + (2.41e8 - 8.78e7i)T + (9.18e16 - 7.70e16i)T^{2} \) |
| 83 | \( 1 + (-8.41e6 - 1.45e7i)T + (-9.34e16 + 1.61e17i)T^{2} \) |
| 89 | \( 1 + (7.68e8 + 2.79e8i)T + (2.68e17 + 2.25e17i)T^{2} \) |
| 97 | \( 1 + (-4.16e8 - 3.49e8i)T + (1.32e17 + 7.48e17i)T^{2} \) |
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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
−13.84344152022767719730458008467, −12.32243053903973835874824733149, −11.87734833038365336294132721227, −10.32380041647911888652117045037, −8.820897655681669847764969833080, −8.303640291469114353731589431800, −5.69969345413350468443391042095, −4.73984147376036422071205555761, −2.19415057436681187600346941351, −0.72271569486083436716662699792,
1.10295435598058224062362600606, 3.28265937734245953602691449962, 5.63170201753921592485673615736, 6.88823767060159378789465602124, 7.74851494454797627085262967039, 9.795928593856745016948023203481, 10.81595708747039704102661699952, 11.61446559561823764965829184872, 13.99228434973898350581886869701, 14.30566520003245971132331771403
