| L(s) = 1 | − 23.5i·2-s + 469.·4-s + 4.42e3i·5-s − 2.15e4·7-s − 3.51e4i·8-s + 1.04e5·10-s − 2.42e5i·11-s − 3.98e5·13-s + 5.07e5i·14-s − 3.46e5·16-s − 2.02e6i·17-s − 2.77e6·19-s + 2.07e6i·20-s − 5.71e6·22-s + 2.47e6i·23-s + ⋯ |
| L(s) = 1 | − 0.735i·2-s + 0.458·4-s + 1.41i·5-s − 1.28·7-s − 1.07i·8-s + 1.04·10-s − 1.50i·11-s − 1.07·13-s + 0.944i·14-s − 0.330·16-s − 1.42i·17-s − 1.11·19-s + 0.649i·20-s − 1.10·22-s + 0.385i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.571440i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.571440i\) |
| \(L(6)\) |
|
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 | \( 1 + 23.5iT - 1.02e3T^{2} \) |
| 5 | \( 1 - 4.42e3iT - 9.76e6T^{2} \) |
| 7 | \( 1 + 2.15e4T + 2.82e8T^{2} \) |
| 11 | \( 1 + 2.42e5iT - 2.59e10T^{2} \) |
| 13 | \( 1 + 3.98e5T + 1.37e11T^{2} \) |
| 17 | \( 1 + 2.02e6iT - 2.01e12T^{2} \) |
| 19 | \( 1 + 2.77e6T + 6.13e12T^{2} \) |
| 23 | \( 1 - 2.47e6iT - 4.14e13T^{2} \) |
| 29 | \( 1 - 9.13e6iT - 4.20e14T^{2} \) |
| 31 | \( 1 + 1.81e7T + 8.19e14T^{2} \) |
| 37 | \( 1 - 1.14e7T + 4.80e15T^{2} \) |
| 41 | \( 1 - 1.37e8iT - 1.34e16T^{2} \) |
| 43 | \( 1 - 9.87e7T + 2.16e16T^{2} \) |
| 47 | \( 1 + 1.41e8iT - 5.25e16T^{2} \) |
| 53 | \( 1 - 1.39e8iT - 1.74e17T^{2} \) |
| 59 | \( 1 + 1.06e9iT - 5.11e17T^{2} \) |
| 61 | \( 1 + 1.44e9T + 7.13e17T^{2} \) |
| 67 | \( 1 - 1.44e9T + 1.82e18T^{2} \) |
| 71 | \( 1 - 1.33e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 + 1.46e9T + 4.29e18T^{2} \) |
| 79 | \( 1 + 4.95e9T + 9.46e18T^{2} \) |
| 83 | \( 1 + 5.60e8iT - 1.55e19T^{2} \) |
| 89 | \( 1 - 4.73e9iT - 3.11e19T^{2} \) |
| 97 | \( 1 - 1.21e10T + 7.37e19T^{2} \) |
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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
−14.31756189032383292932337384882, −12.96253613513396274214763937000, −11.55866840452910814422357556719, −10.63811767767879850371030810109, −9.558558894065730814420772934894, −7.18573647926750931229865690052, −6.24730850021063798243171825322, −3.32622235007718806256692791193, −2.63688459425312409551123160719, −0.19448467961833069147089120604,
2.04851379550250701695805129615, 4.50422288553199858725295390026, 6.04911401234464673622362401626, 7.39299067325464847989275181256, 8.877162990224596184619576505148, 10.19930954679688831124228614533, 12.37551980330806409485231484260, 12.78114075617125698737729912213, 14.80247264507891500750295138171, 15.70685744998986674596597434739
