| L(s) = 1 | + (−4.57 − 1.66i)2-s + (−24.3 − 138. i)3-s + (−374. − 313. i)4-s + (461. − 2.61e3i)5-s + (−118. + 672. i)6-s + (5.32e3 − 4.46e3i)7-s + (2.43e3 + 4.21e3i)8-s + (−1.84e4 + 6.73e3i)9-s + (−6.47e3 + 1.12e4i)10-s + (−5.91e3 − 3.35e4i)11-s + (−3.42e4 + 5.93e4i)12-s + (1.15e5 − 4.21e4i)13-s + (−3.17e4 + 1.15e4i)14-s + (−3.72e5 − 13.7i)15-s + (3.92e4 + 2.22e5i)16-s + (5.48e4 − 9.50e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.202 − 0.0736i)2-s + (−0.173 − 0.984i)3-s + (−0.730 − 0.613i)4-s + (0.330 − 1.87i)5-s + (−0.0373 + 0.211i)6-s + (0.837 − 0.702i)7-s + (0.210 + 0.364i)8-s + (−0.939 + 0.341i)9-s + (−0.204 + 0.354i)10-s + (−0.121 − 0.690i)11-s + (−0.476 + 0.825i)12-s + (1.12 − 0.409i)13-s + (−0.221 + 0.0804i)14-s + (−1.90 − 7.02e−5i)15-s + (0.149 + 0.850i)16-s + (0.159 − 0.275i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.286i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.957 - 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.207608 + 1.41678i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.207608 + 1.41678i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (24.3 + 138. i)T \) |
| good | 2 | \( 1 + (4.57 + 1.66i)T + (392. + 329. i)T^{2} \) |
| 5 | \( 1 + (-461. + 2.61e3i)T + (-1.83e6 - 6.68e5i)T^{2} \) |
| 7 | \( 1 + (-5.32e3 + 4.46e3i)T + (7.00e6 - 3.97e7i)T^{2} \) |
| 11 | \( 1 + (5.91e3 + 3.35e4i)T + (-2.21e9 + 8.06e8i)T^{2} \) |
| 13 | \( 1 + (-1.15e5 + 4.21e4i)T + (8.12e9 - 6.81e9i)T^{2} \) |
| 17 | \( 1 + (-5.48e4 + 9.50e4i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 19 | \( 1 + (2.96e3 + 5.13e3i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (-1.40e5 - 1.18e5i)T + (3.12e11 + 1.77e12i)T^{2} \) |
| 29 | \( 1 + (-5.64e6 - 2.05e6i)T + (1.11e13 + 9.32e12i)T^{2} \) |
| 31 | \( 1 + (-6.88e6 - 5.77e6i)T + (4.59e12 + 2.60e13i)T^{2} \) |
| 37 | \( 1 + (2.83e6 - 4.91e6i)T + (-6.49e13 - 1.12e14i)T^{2} \) |
| 41 | \( 1 + (-3.13e7 + 1.14e7i)T + (2.50e14 - 2.10e14i)T^{2} \) |
| 43 | \( 1 + (1.71e6 + 9.71e6i)T + (-4.72e14 + 1.71e14i)T^{2} \) |
| 47 | \( 1 + (-2.21e7 + 1.85e7i)T + (1.94e14 - 1.10e15i)T^{2} \) |
| 53 | \( 1 + 1.54e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + (1.95e6 - 1.10e7i)T + (-8.14e15 - 2.96e15i)T^{2} \) |
| 61 | \( 1 + (7.31e7 - 6.13e7i)T + (2.03e15 - 1.15e16i)T^{2} \) |
| 67 | \( 1 + (2.31e8 - 8.44e7i)T + (2.08e16 - 1.74e16i)T^{2} \) |
| 71 | \( 1 + (-6.37e7 + 1.10e8i)T + (-2.29e16 - 3.97e16i)T^{2} \) |
| 73 | \( 1 + (-1.66e7 - 2.87e7i)T + (-2.94e16 + 5.09e16i)T^{2} \) |
| 79 | \( 1 + (7.97e7 + 2.90e7i)T + (9.18e16 + 7.70e16i)T^{2} \) |
| 83 | \( 1 + (-1.82e8 - 6.62e7i)T + (1.43e17 + 1.20e17i)T^{2} \) |
| 89 | \( 1 + (-2.53e8 - 4.38e8i)T + (-1.75e17 + 3.03e17i)T^{2} \) |
| 97 | \( 1 + (9.83e7 + 5.57e8i)T + (-7.14e17 + 2.60e17i)T^{2} \) |
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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
−13.88785706894380663366458275199, −13.53649050633446451920371025272, −12.23967724161621190478268125585, −10.67424868156762479714971505140, −8.804633972916154814418068405962, −8.185261428514399982025263334571, −5.79858951097892386643798063061, −4.70475874375093643991056989622, −1.21519488790610224345513913966, −0.839730690171551059416666238313,
2.76922501312437855909826842124, 4.28863420043497006810164067279, 6.16972277603921365862844277327, 8.028819287893640210994673115942, 9.504595395875245992519185895865, 10.66880774506795324793007749316, 11.76402451075383688329160296761, 13.85096850323689092965937570894, 14.74068364410607695873163250688, 15.73993032669501777972564707136
