| L(s) = 1 | + (1.48e3 − 2.58e3i)2-s + (−2.42e4 − 3.05e5i)3-s + (−2.43e5 − 4.21e5i)4-s + (−6.02e7 − 1.04e8i)5-s + (−8.25e8 − 3.92e8i)6-s + (4.52e9 − 7.83e9i)7-s + (2.35e10 − 1.78e−6i)8-s + (−9.29e10 + 1.48e10i)9-s − 3.59e11·10-s + (5.25e11 − 9.09e11i)11-s + (−1.23e11 + 8.47e10i)12-s + (−3.79e11 − 6.56e11i)13-s + (−1.34e13 − 2.33e13i)14-s + (−3.04e13 + 2.09e13i)15-s + (3.71e13 − 6.42e13i)16-s − 1.03e14·17-s + ⋯ |
| L(s) = 1 | + (0.514 − 0.890i)2-s + (−0.0791 − 0.996i)3-s + (−0.0290 − 0.0502i)4-s + (−0.551 − 0.955i)5-s + (−0.928 − 0.442i)6-s + (0.865 − 1.49i)7-s + 0.968·8-s + (−0.987 + 0.157i)9-s − 1.13·10-s + (0.554 − 0.960i)11-s + (−0.0478 + 0.0329i)12-s + (−0.0587 − 0.101i)13-s + (−0.889 − 1.54i)14-s + (−0.909 + 0.625i)15-s + (0.527 − 0.913i)16-s − 0.735·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.857 - 0.513i)\, \overline{\Lambda}(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & (-0.857 - 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(12)\) |
\(\approx\) |
\(2.651325007\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.651325007\) |
| \(L(\frac{25}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.42e4 + 3.05e5i)T \) |
| good | 2 | \( 1 + (-1.48e3 + 2.58e3i)T + (-4.19e6 - 7.26e6i)T^{2} \) |
| 5 | \( 1 + (6.02e7 + 1.04e8i)T + (-5.96e15 + 1.03e16i)T^{2} \) |
| 7 | \( 1 + (-4.52e9 + 7.83e9i)T + (-1.36e19 - 2.37e19i)T^{2} \) |
| 11 | \( 1 + (-5.25e11 + 9.09e11i)T + (-4.47e23 - 7.75e23i)T^{2} \) |
| 13 | \( 1 + (3.79e11 + 6.56e11i)T + (-2.08e25 + 3.61e25i)T^{2} \) |
| 17 | \( 1 + 1.03e14T + 1.99e28T^{2} \) |
| 19 | \( 1 - 1.80e14T + 2.57e29T^{2} \) |
| 23 | \( 1 + (-3.56e15 - 6.16e15i)T + (-1.04e31 + 1.80e31i)T^{2} \) |
| 29 | \( 1 + (-2.37e15 + 4.10e15i)T + (-2.15e33 - 3.73e33i)T^{2} \) |
| 31 | \( 1 + (-4.85e16 - 8.40e16i)T + (-1.00e34 + 1.73e34i)T^{2} \) |
| 37 | \( 1 + 1.78e18T + 1.17e36T^{2} \) |
| 41 | \( 1 + (-3.04e18 - 5.27e18i)T + (-6.20e36 + 1.07e37i)T^{2} \) |
| 43 | \( 1 + (2.47e18 - 4.28e18i)T + (-1.85e37 - 3.21e37i)T^{2} \) |
| 47 | \( 1 + (-6.45e18 + 1.11e19i)T + (-1.43e38 - 2.48e38i)T^{2} \) |
| 53 | \( 1 - 6.83e19T + 4.55e39T^{2} \) |
| 59 | \( 1 + (-8.75e19 - 1.51e20i)T + (-2.68e40 + 4.64e40i)T^{2} \) |
| 61 | \( 1 + (-2.87e20 + 4.97e20i)T + (-5.77e40 - 1.00e41i)T^{2} \) |
| 67 | \( 1 + (7.92e20 + 1.37e21i)T + (-4.99e41 + 8.65e41i)T^{2} \) |
| 71 | \( 1 + 3.59e19T + 3.79e42T^{2} \) |
| 73 | \( 1 + 1.14e21T + 7.18e42T^{2} \) |
| 79 | \( 1 + (-1.62e20 + 2.81e20i)T + (-2.21e43 - 3.82e43i)T^{2} \) |
| 83 | \( 1 + (1.09e21 - 1.89e21i)T + (-6.88e43 - 1.19e44i)T^{2} \) |
| 89 | \( 1 + 1.02e22T + 6.85e44T^{2} \) |
| 97 | \( 1 + (-4.87e22 + 8.43e22i)T + (-2.48e45 - 4.29e45i)T^{2} \) |
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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
−13.95270792445413169865678304908, −13.19373524012603596159076481159, −11.80538047622396404123917183173, −11.01080655799766088701558340611, −8.346606212069443311359860984831, −7.20963555737514418541089554588, −4.81974242793594471608817823272, −3.48019174658288225334723039305, −1.47017114430992251550762530890, −0.76500442819671645554011621248,
2.36025348330142916824792541459, 4.31023116360241014339425359373, 5.46640494901548677809625443040, 6.95873878546889449712821044840, 8.767984292812582861116215079568, 10.63590923020150733315783254452, 11.84484256023470129933313798941, 14.47635675528188047035731148175, 15.02248742309095537077247370194, 15.76808785854372989868184068616
