L(s) = 1 | + (12.2 + 7.06i)2-s + (67.7 + 117. i)4-s + (103. − 59.9i)5-s + (−128. + 223. i)7-s + 1.00e3i·8-s + 1.69e3·10-s + (−518. − 299. i)11-s + (−968. − 1.67e3i)13-s + (−3.15e3 + 1.82e3i)14-s + (−2.79e3 + 4.84e3i)16-s − 4.87e3i·17-s + 7.68e3·19-s + (1.40e4 + 8.12e3i)20-s + (−4.22e3 − 7.32e3i)22-s + (594. − 343. i)23-s + ⋯ |
L(s) = 1 | + (1.52 + 0.882i)2-s + (1.05 + 1.83i)4-s + (0.830 − 0.479i)5-s + (−0.375 + 0.650i)7-s + 1.97i·8-s + 1.69·10-s + (−0.389 − 0.224i)11-s + (−0.440 − 0.763i)13-s + (−1.14 + 0.663i)14-s + (−0.682 + 1.18i)16-s − 0.992i·17-s + 1.12·19-s + (1.75 + 1.01i)20-s + (−0.397 − 0.687i)22-s + (0.0488 − 0.0282i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.329 - 0.944i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.329 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(2.97978 + 2.11523i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.97978 + 2.11523i\) |
\(L(4)\) |
|
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 | \( 1 + (-12.2 - 7.06i)T + (32 + 55.4i)T^{2} \) |
| 5 | \( 1 + (-103. + 59.9i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 7 | \( 1 + (128. - 223. i)T + (-5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (518. + 299. i)T + (8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 + (968. + 1.67e3i)T + (-2.41e6 + 4.18e6i)T^{2} \) |
| 17 | \( 1 + 4.87e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 7.68e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + (-594. + 343. i)T + (7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 + (2.99e4 + 1.73e4i)T + (2.97e8 + 5.15e8i)T^{2} \) |
| 31 | \( 1 + (1.38e3 + 2.40e3i)T + (-4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 - 1.71e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + (7.15e4 - 4.13e4i)T + (2.37e9 - 4.11e9i)T^{2} \) |
| 43 | \( 1 + (6.32e4 - 1.09e5i)T + (-3.16e9 - 5.47e9i)T^{2} \) |
| 47 | \( 1 + (-7.20e4 - 4.15e4i)T + (5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 - 4.00e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 + (-5.76e4 + 3.32e4i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (1.84e5 - 3.19e5i)T + (-2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (1.27e5 + 2.21e5i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 - 2.69e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 5.51e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + (-3.55e5 + 6.14e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + (3.01e5 + 1.74e5i)T + (1.63e11 + 2.83e11i)T^{2} \) |
| 89 | \( 1 + 8.36e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (4.70e4 - 8.14e4i)T + (-4.16e11 - 7.21e11i)T^{2} \) |
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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
−15.96586322102358318666612622935, −14.99464795670260556896308797850, −13.67242647787594667684135245583, −12.98384112448284244301202818961, −11.78612428192529888161875800963, −9.520039371731718702757019757691, −7.61023053184990458951289261054, −5.92751989994942189505152341744, −5.08471949449368559494640849524, −2.92003064202040674685120633271,
1.98235432910698225670101650146, 3.65702259340915853468927871094, 5.38092182516226455454686350498, 6.83000615608205668523734305530, 9.841408692747138538782020713016, 10.79283520160082938376379525523, 12.19523753356736156012501509200, 13.40240872460601434144562715745, 14.08795653766865631676617574139, 15.21745865237627371767158767527