| L(s) = 1 | + (2.50e3 − 4.33e3i)2-s + (1.51e5 − 2.66e5i)3-s + (−8.33e6 − 1.44e7i)4-s + (−5.32e6 − 9.23e6i)5-s + (−7.75e8 − 1.32e9i)6-s + (−2.85e9 + 4.94e9i)7-s + (−4.14e10 − 7.62e−6i)8-s + (−4.79e10 − 8.09e10i)9-s − 5.33e10·10-s + (−5.20e11 + 9.01e11i)11-s + (−5.11e12 + 2.89e10i)12-s + (3.60e11 + 6.23e11i)13-s + (1.43e13 + 2.47e13i)14-s + (−3.27e12 + 1.85e10i)15-s + (−3.39e13 + 5.87e13i)16-s − 1.07e14·17-s + ⋯ |
| L(s) = 1 | + (0.864 − 1.49i)2-s + (0.495 − 0.868i)3-s + (−0.993 − 1.72i)4-s + (−0.0488 − 0.0845i)5-s + (−0.872 − 1.49i)6-s + (−0.546 + 0.945i)7-s − 1.70·8-s + (−0.509 − 0.860i)9-s − 0.168·10-s + (−0.550 + 0.952i)11-s + (−1.98 + 0.0112i)12-s + (0.0557 + 0.0965i)13-s + (0.943 + 1.63i)14-s + (−0.0976 + 0.000552i)15-s + (−0.481 + 0.834i)16-s − 0.763·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.162 - 0.986i)\, \overline{\Lambda}(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & (0.162 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(12)\) |
\(\approx\) |
\(1.392134211\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.392134211\) |
| \(L(\frac{25}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.51e5 + 2.66e5i)T \) |
| good | 2 | \( 1 + (-2.50e3 + 4.33e3i)T + (-4.19e6 - 7.26e6i)T^{2} \) |
| 5 | \( 1 + (5.32e6 + 9.23e6i)T + (-5.96e15 + 1.03e16i)T^{2} \) |
| 7 | \( 1 + (2.85e9 - 4.94e9i)T + (-1.36e19 - 2.37e19i)T^{2} \) |
| 11 | \( 1 + (5.20e11 - 9.01e11i)T + (-4.47e23 - 7.75e23i)T^{2} \) |
| 13 | \( 1 + (-3.60e11 - 6.23e11i)T + (-2.08e25 + 3.61e25i)T^{2} \) |
| 17 | \( 1 + 1.07e14T + 1.99e28T^{2} \) |
| 19 | \( 1 + 5.15e14T + 2.57e29T^{2} \) |
| 23 | \( 1 + (3.82e15 + 6.63e15i)T + (-1.04e31 + 1.80e31i)T^{2} \) |
| 29 | \( 1 + (-4.99e16 + 8.65e16i)T + (-2.15e33 - 3.73e33i)T^{2} \) |
| 31 | \( 1 + (-8.59e16 - 1.48e17i)T + (-1.00e34 + 1.73e34i)T^{2} \) |
| 37 | \( 1 + 1.40e18T + 1.17e36T^{2} \) |
| 41 | \( 1 + (-1.97e18 - 3.42e18i)T + (-6.20e36 + 1.07e37i)T^{2} \) |
| 43 | \( 1 + (-2.78e18 + 4.82e18i)T + (-1.85e37 - 3.21e37i)T^{2} \) |
| 47 | \( 1 + (-7.36e18 + 1.27e19i)T + (-1.43e38 - 2.48e38i)T^{2} \) |
| 53 | \( 1 - 5.78e19T + 4.55e39T^{2} \) |
| 59 | \( 1 + (2.07e20 + 3.58e20i)T + (-2.68e40 + 4.64e40i)T^{2} \) |
| 61 | \( 1 + (1.86e20 - 3.22e20i)T + (-5.77e40 - 1.00e41i)T^{2} \) |
| 67 | \( 1 + (5.55e20 + 9.62e20i)T + (-4.99e41 + 8.65e41i)T^{2} \) |
| 71 | \( 1 + 1.08e21T + 3.79e42T^{2} \) |
| 73 | \( 1 + 8.96e20T + 7.18e42T^{2} \) |
| 79 | \( 1 + (-1.14e21 + 1.97e21i)T + (-2.21e43 - 3.82e43i)T^{2} \) |
| 83 | \( 1 + (6.44e21 - 1.11e22i)T + (-6.88e43 - 1.19e44i)T^{2} \) |
| 89 | \( 1 - 1.03e21T + 6.85e44T^{2} \) |
| 97 | \( 1 + (2.37e22 - 4.11e22i)T + (-2.48e45 - 4.29e45i)T^{2} \) |
| 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
−13.90959979072437627463220716341, −12.61360013432492329231372133205, −12.15653006530089004353795775624, −10.28205714969518737713053680798, −8.678382508422493548407973249830, −6.32812815075361908379220553499, −4.45268649472352144307971529129, −2.67947823195296311222657799056, −2.06715760792385768883709338573, −0.28251439467456063453457771878,
3.28481469094905394343423981903, 4.34495862086530707939280538736, 5.80925754983497082493526751973, 7.38255974932621287751341400918, 8.738816609705818076843371344633, 10.64296730621229620940045892978, 13.24667004594191291599146412241, 14.01600473001461614966769012538, 15.39281303966207634388942781517, 16.20354475256052714678158499492
