| L(s) = 1 | + (−51.7 − 37.6i)2-s + (226. − 696. i)3-s + (1.26e3 + 3.89e3i)4-s + (3.95e4 − 2.87e4i)5-s + (−3.79e4 + 2.75e4i)6-s + (−1.00e5 − 3.09e5i)7-s + (8.10e4 − 2.49e5i)8-s + (8.55e5 + 6.21e5i)9-s − 3.12e6·10-s + (5.86e6 − 2.64e5i)11-s + 2.99e6·12-s + (1.80e7 + 1.30e7i)13-s + (−6.42e6 + 1.97e7i)14-s + (−1.10e7 − 3.40e7i)15-s + (−1.35e7 + 9.86e6i)16-s + (1.28e8 − 9.30e7i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 − 0.415i)2-s + (0.179 − 0.551i)3-s + (0.154 + 0.475i)4-s + (1.13 − 0.821i)5-s + (−0.331 + 0.241i)6-s + (−0.322 − 0.993i)7-s + (0.109 − 0.336i)8-s + (0.536 + 0.390i)9-s − 0.988·10-s + (0.998 − 0.0449i)11-s + 0.289·12-s + (1.03 + 0.752i)13-s + (−0.228 + 0.702i)14-s + (−0.250 − 0.770i)15-s + (−0.202 + 0.146i)16-s + (1.28 − 0.934i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.242 + 0.970i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.242 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(2.147805526\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.147805526\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (51.7 + 37.6i)T \) |
| 11 | \( 1 + (-5.86e6 + 2.64e5i)T \) |
| good | 3 | \( 1 + (-226. + 696. i)T + (-1.28e6 - 9.37e5i)T^{2} \) |
| 5 | \( 1 + (-3.95e4 + 2.87e4i)T + (3.77e8 - 1.16e9i)T^{2} \) |
| 7 | \( 1 + (1.00e5 + 3.09e5i)T + (-7.83e10 + 5.69e10i)T^{2} \) |
| 13 | \( 1 + (-1.80e7 - 1.30e7i)T + (9.35e13 + 2.88e14i)T^{2} \) |
| 17 | \( 1 + (-1.28e8 + 9.30e7i)T + (3.06e15 - 9.41e15i)T^{2} \) |
| 19 | \( 1 + (8.70e7 - 2.67e8i)T + (-3.40e16 - 2.47e16i)T^{2} \) |
| 23 | \( 1 + 2.29e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + (1.72e9 + 5.30e9i)T + (-8.30e18 + 6.03e18i)T^{2} \) |
| 31 | \( 1 + (3.49e9 + 2.53e9i)T + (7.54e18 + 2.32e19i)T^{2} \) |
| 37 | \( 1 + (-6.21e9 - 1.91e10i)T + (-1.97e20 + 1.43e20i)T^{2} \) |
| 41 | \( 1 + (-1.50e10 + 4.64e10i)T + (-7.48e20 - 5.43e20i)T^{2} \) |
| 43 | \( 1 + 4.19e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + (2.72e10 - 8.38e10i)T + (-4.41e21 - 3.20e21i)T^{2} \) |
| 53 | \( 1 + (-3.94e10 - 2.86e10i)T + (8.04e21 + 2.47e22i)T^{2} \) |
| 59 | \( 1 + (-4.38e10 - 1.35e11i)T + (-8.49e22 + 6.17e22i)T^{2} \) |
| 61 | \( 1 + (5.18e11 - 3.76e11i)T + (5.00e22 - 1.53e23i)T^{2} \) |
| 67 | \( 1 + 5.67e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + (-5.56e11 + 4.04e11i)T + (3.60e23 - 1.10e24i)T^{2} \) |
| 73 | \( 1 + (6.46e11 + 1.99e12i)T + (-1.35e24 + 9.82e23i)T^{2} \) |
| 79 | \( 1 + (-1.85e12 - 1.34e12i)T + (1.44e24 + 4.43e24i)T^{2} \) |
| 83 | \( 1 + (-6.35e9 + 4.61e9i)T + (2.74e24 - 8.43e24i)T^{2} \) |
| 89 | \( 1 + 3.84e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + (-1.37e12 - 9.97e11i)T + (2.07e25 + 6.40e25i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.95337761695110061131992215905, −13.32224254990919248343034790262, −11.98895386927057480587446240332, −10.21264560846698415838714757922, −9.281688312973286892537280872226, −7.72191224505332936984782096414, −6.20055445008397874044662176655, −4.00120065659797699646655866854, −1.74000834524727526038008862267, −1.02315070421708277532816146290,
1.52141985936689446424554174768, 3.28874827026046831584503927068, 5.69814175175048880307911196567, 6.69046082267012231875524793250, 8.815472043949775927976959388888, 9.731400164041372099087501242183, 10.82552154284785737203575789325, 12.78683571864733240512412319051, 14.48528847627938032855488096787, 15.18845329466982286257429191319
