L(s) = 1 | − 8.26i·2-s + 955.·4-s − 2.84e3i·5-s + 2.96e3·7-s − 1.63e4i·8-s − 2.34e4·10-s + 5.70e4i·11-s + 1.76e5·13-s − 2.44e4i·14-s + 8.43e5·16-s − 2.37e6i·17-s − 2.78e6·19-s − 2.71e6i·20-s + 4.71e5·22-s − 6.45e6i·23-s + ⋯ |
L(s) = 1 | − 0.258i·2-s + 0.933·4-s − 0.909i·5-s + 0.176·7-s − 0.499i·8-s − 0.234·10-s + 0.354i·11-s + 0.474·13-s − 0.0455i·14-s + 0.804·16-s − 1.67i·17-s − 1.12·19-s − 0.848i·20-s + 0.0915·22-s − 1.00i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{11}{2})\) |
\(\approx\) |
\(1.59000 - 1.59000i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.59000 - 1.59000i\) |
\(L(6)\) |
|
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 | \( 1 + 8.26iT - 1.02e3T^{2} \) |
| 5 | \( 1 + 2.84e3iT - 9.76e6T^{2} \) |
| 7 | \( 1 - 2.96e3T + 2.82e8T^{2} \) |
| 11 | \( 1 - 5.70e4iT - 2.59e10T^{2} \) |
| 13 | \( 1 - 1.76e5T + 1.37e11T^{2} \) |
| 17 | \( 1 + 2.37e6iT - 2.01e12T^{2} \) |
| 19 | \( 1 + 2.78e6T + 6.13e12T^{2} \) |
| 23 | \( 1 + 6.45e6iT - 4.14e13T^{2} \) |
| 29 | \( 1 + 3.39e7iT - 4.20e14T^{2} \) |
| 31 | \( 1 + 1.12e7T + 8.19e14T^{2} \) |
| 37 | \( 1 - 9.01e7T + 4.80e15T^{2} \) |
| 41 | \( 1 - 1.73e8iT - 1.34e16T^{2} \) |
| 43 | \( 1 - 4.56e7T + 2.16e16T^{2} \) |
| 47 | \( 1 - 1.90e8iT - 5.25e16T^{2} \) |
| 53 | \( 1 - 3.24e8iT - 1.74e17T^{2} \) |
| 59 | \( 1 - 4.01e8iT - 5.11e17T^{2} \) |
| 61 | \( 1 - 6.57e8T + 7.13e17T^{2} \) |
| 67 | \( 1 + 2.39e9T + 1.82e18T^{2} \) |
| 71 | \( 1 + 2.19e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 - 3.96e9T + 4.29e18T^{2} \) |
| 79 | \( 1 - 7.14e8T + 9.46e18T^{2} \) |
| 83 | \( 1 - 3.95e9iT - 1.55e19T^{2} \) |
| 89 | \( 1 - 5.68e9iT - 3.11e19T^{2} \) |
| 97 | \( 1 - 8.34e9T + 7.37e19T^{2} \) |
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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
−14.84859027049322452442991577523, −13.18361654899534436800949421481, −12.09369831178910831055144497109, −10.98970419864718575584372228253, −9.471612999230756786288659057822, −7.919822362842997365752966416618, −6.34540470756886862516188759968, −4.55759967120560174324048881241, −2.48412521665356200993173037815, −0.871457218913790271055644022558,
1.83074392813609514746081481660, 3.45884176110423094779366280872, 5.89517241815391602445192784167, 6.97913634080153075790242860904, 8.400801377214452573785078445942, 10.53401866580336478781670875954, 11.18444341466619832767840966596, 12.76749018869906039949585630329, 14.47752307896603179065022730259, 15.18671641229164150619509149324