| L(s) = 1 | + (12.1 − 7.01i)2-s + (−29.5 + 51.1i)4-s + (−676. − 390. i)5-s + (2.16e3 + 3.75e3i)7-s + 4.42e3i·8-s − 1.09e4·10-s + (−5.60e3 + 3.23e3i)11-s + (−8.49e3 + 1.47e4i)13-s + (5.27e4 + 3.04e4i)14-s + (2.34e4 + 4.06e4i)16-s + 2.98e4i·17-s − 9.17e4·19-s + (3.99e4 − 2.30e4i)20-s + (−4.54e4 + 7.86e4i)22-s + (1.06e5 + 6.14e4i)23-s + ⋯ |
| L(s) = 1 | + (0.759 − 0.438i)2-s + (−0.115 + 0.199i)4-s + (−1.08 − 0.624i)5-s + (0.903 + 1.56i)7-s + 1.07i·8-s − 1.09·10-s + (−0.382 + 0.221i)11-s + (−0.297 + 0.515i)13-s + (1.37 + 0.792i)14-s + (0.358 + 0.620i)16-s + 0.357i·17-s − 0.704·19-s + (0.249 − 0.144i)20-s + (−0.193 + 0.335i)22-s + (0.380 + 0.219i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0371 - 0.999i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.0371 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.13587 + 1.09445i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.13587 + 1.09445i\) |
| \(L(5)\) |
|
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 | \( 1 + (-12.1 + 7.01i)T + (128 - 221. i)T^{2} \) |
| 5 | \( 1 + (676. + 390. i)T + (1.95e5 + 3.38e5i)T^{2} \) |
| 7 | \( 1 + (-2.16e3 - 3.75e3i)T + (-2.88e6 + 4.99e6i)T^{2} \) |
| 11 | \( 1 + (5.60e3 - 3.23e3i)T + (1.07e8 - 1.85e8i)T^{2} \) |
| 13 | \( 1 + (8.49e3 - 1.47e4i)T + (-4.07e8 - 7.06e8i)T^{2} \) |
| 17 | \( 1 - 2.98e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 9.17e4T + 1.69e10T^{2} \) |
| 23 | \( 1 + (-1.06e5 - 6.14e4i)T + (3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 + (4.34e5 - 2.50e5i)T + (2.50e11 - 4.33e11i)T^{2} \) |
| 31 | \( 1 + (-5.05e5 + 8.76e5i)T + (-4.26e11 - 7.38e11i)T^{2} \) |
| 37 | \( 1 - 1.90e5T + 3.51e12T^{2} \) |
| 41 | \( 1 + (-1.95e6 - 1.12e6i)T + (3.99e12 + 6.91e12i)T^{2} \) |
| 43 | \( 1 + (1.64e6 + 2.84e6i)T + (-5.84e12 + 1.01e13i)T^{2} \) |
| 47 | \( 1 + (-1.10e5 + 6.36e4i)T + (1.19e13 - 2.06e13i)T^{2} \) |
| 53 | \( 1 - 1.40e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + (-1.79e7 - 1.03e7i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (-9.62e6 - 1.66e7i)T + (-9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-1.17e7 + 2.02e7i)T + (-2.03e14 - 3.51e14i)T^{2} \) |
| 71 | \( 1 - 9.12e6iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 1.43e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + (-2.69e7 - 4.67e7i)T + (-7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 + (2.66e7 - 1.53e7i)T + (1.12e15 - 1.95e15i)T^{2} \) |
| 89 | \( 1 + 2.32e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + (-2.46e7 - 4.27e7i)T + (-3.91e15 + 6.78e15i)T^{2} \) |
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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
−15.46838162998895135834271599056, −14.64650295499443296874337166233, −12.93331505047194496263363453759, −12.04669221754078793273896968420, −11.35668395013062171048213647499, −8.839585741113729762324759236192, −7.979394298870688996405152686912, −5.33003983708554429985823374246, −4.21073194482466840688888147862, −2.29265897429407024200699305851,
0.57057402311931703886257206835, 3.70097276673482717403343432271, 4.85391055437550160796667292838, 6.89298930275634526097002644326, 7.906393648803181141193572388377, 10.29388938867806261918620500740, 11.25752320470857406962082728311, 13.01848376180325605862615418800, 14.20414824233821954435945159788, 14.94661037256003291573534696793
