| L(s) = 1 | + (168. − 292. i)2-s + (−4.05e4 − 7.02e4i)4-s + (1.18e5 + 2.05e5i)5-s + (−1.07e6 + 1.86e6i)7-s − 1.63e7·8-s + 8.01e7·10-s + (6.83e6 − 1.18e7i)11-s + (8.98e7 + 1.55e8i)13-s + (3.63e8 + 6.29e8i)14-s + (−1.42e9 + 2.46e9i)16-s + 6.86e8·17-s + 6.26e9·19-s + (9.62e9 − 1.66e10i)20-s + (−2.30e9 − 3.99e9i)22-s + (3.71e9 + 6.42e9i)23-s + ⋯ |
| L(s) = 1 | + (0.931 − 1.61i)2-s + (−1.23 − 2.14i)4-s + (0.679 + 1.17i)5-s + (−0.493 + 0.855i)7-s − 2.74·8-s + 2.53·10-s + (0.105 − 0.183i)11-s + (0.397 + 0.687i)13-s + (0.920 + 1.59i)14-s + (−1.32 + 2.29i)16-s + 0.405·17-s + 1.60·19-s + (1.68 − 2.91i)20-s + (−0.197 − 0.341i)22-s + (0.227 + 0.393i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.692 + 0.721i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (0.692 + 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(3.233975589\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.233975589\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (-168. + 292. i)T + (-1.63e4 - 2.83e4i)T^{2} \) |
| 5 | \( 1 + (-1.18e5 - 2.05e5i)T + (-1.52e10 + 2.64e10i)T^{2} \) |
| 7 | \( 1 + (1.07e6 - 1.86e6i)T + (-2.37e12 - 4.11e12i)T^{2} \) |
| 11 | \( 1 + (-6.83e6 + 1.18e7i)T + (-2.08e15 - 3.61e15i)T^{2} \) |
| 13 | \( 1 + (-8.98e7 - 1.55e8i)T + (-2.55e16 + 4.43e16i)T^{2} \) |
| 17 | \( 1 - 6.86e8T + 2.86e18T^{2} \) |
| 19 | \( 1 - 6.26e9T + 1.51e19T^{2} \) |
| 23 | \( 1 + (-3.71e9 - 6.42e9i)T + (-1.33e20 + 2.30e20i)T^{2} \) |
| 29 | \( 1 + (-2.15e10 + 3.72e10i)T + (-4.31e21 - 7.47e21i)T^{2} \) |
| 31 | \( 1 + (-1.30e11 - 2.25e11i)T + (-1.17e22 + 2.03e22i)T^{2} \) |
| 37 | \( 1 + 5.98e11T + 3.33e23T^{2} \) |
| 41 | \( 1 + (1.39e10 + 2.41e10i)T + (-7.77e23 + 1.34e24i)T^{2} \) |
| 43 | \( 1 + (1.94e11 - 3.36e11i)T + (-1.58e24 - 2.75e24i)T^{2} \) |
| 47 | \( 1 + (2.11e12 - 3.66e12i)T + (-6.03e24 - 1.04e25i)T^{2} \) |
| 53 | \( 1 - 6.23e12T + 7.31e25T^{2} \) |
| 59 | \( 1 + (8.43e12 + 1.46e13i)T + (-1.82e26 + 3.16e26i)T^{2} \) |
| 61 | \( 1 + (4.37e12 - 7.57e12i)T + (-3.01e26 - 5.21e26i)T^{2} \) |
| 67 | \( 1 + (-5.15e12 - 8.92e12i)T + (-1.23e27 + 2.13e27i)T^{2} \) |
| 71 | \( 1 - 1.64e13T + 5.87e27T^{2} \) |
| 73 | \( 1 - 1.67e14T + 8.90e27T^{2} \) |
| 79 | \( 1 + (8.94e13 - 1.54e14i)T + (-1.45e28 - 2.52e28i)T^{2} \) |
| 83 | \( 1 + (-1.24e14 + 2.15e14i)T + (-3.05e28 - 5.29e28i)T^{2} \) |
| 89 | \( 1 + 1.95e14T + 1.74e29T^{2} \) |
| 97 | \( 1 + (6.24e14 - 1.08e15i)T + (-3.16e29 - 5.48e29i)T^{2} \) |
| 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
−13.73258321681244595895819970578, −12.29103462990442301223480259745, −11.36327001869464656377391651578, −10.19914491954549506635760963669, −9.287834458058505005237649496734, −6.43740105936507501561471569452, −5.28514367601611692509419755273, −3.40383369849489084969802147125, −2.65940202986987005091521086247, −1.35801647966164342957483980000,
0.75684335709927626284772088751, 3.51322245158269079831813051968, 4.87580773192214688451833008028, 5.80623992785535321631450796747, 7.16206775529093988331514248282, 8.398001367133658744495385879328, 9.741108161719844079791498047722, 12.28814294166216211617175187124, 13.33459890192844085459210558682, 13.84981056996399358164839880276
