| L(s) = 1 | + 43.0i·2-s − 829.·4-s + 2.15e3i·5-s − 1.12e4·7-s + 8.36e3i·8-s − 9.27e4·10-s − 9.80e4i·11-s − 5.93e5·13-s − 4.86e5i·14-s − 1.20e6·16-s − 1.77e5i·17-s + 2.84e6·19-s − 1.78e6i·20-s + 4.22e6·22-s − 1.06e7i·23-s + ⋯ |
| L(s) = 1 | + 1.34i·2-s − 0.810·4-s + 0.689i·5-s − 0.671·7-s + 0.255i·8-s − 0.927·10-s − 0.608i·11-s − 1.59·13-s − 0.903i·14-s − 1.15·16-s − 0.124i·17-s + 1.14·19-s − 0.558i·20-s + 0.818·22-s − 1.64i·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
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.269359 - 0.269359i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.269359 - 0.269359i\) |
| \(L(6)\) |
|
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 | \( 1 - 43.0iT - 1.02e3T^{2} \) |
| 5 | \( 1 - 2.15e3iT - 9.76e6T^{2} \) |
| 7 | \( 1 + 1.12e4T + 2.82e8T^{2} \) |
| 11 | \( 1 + 9.80e4iT - 2.59e10T^{2} \) |
| 13 | \( 1 + 5.93e5T + 1.37e11T^{2} \) |
| 17 | \( 1 + 1.77e5iT - 2.01e12T^{2} \) |
| 19 | \( 1 - 2.84e6T + 6.13e12T^{2} \) |
| 23 | \( 1 + 1.06e7iT - 4.14e13T^{2} \) |
| 29 | \( 1 - 2.16e7iT - 4.20e14T^{2} \) |
| 31 | \( 1 + 2.33e7T + 8.19e14T^{2} \) |
| 37 | \( 1 + 1.23e8T + 4.80e15T^{2} \) |
| 41 | \( 1 + 9.53e7iT - 1.34e16T^{2} \) |
| 43 | \( 1 + 3.63e7T + 2.16e16T^{2} \) |
| 47 | \( 1 - 4.01e8iT - 5.25e16T^{2} \) |
| 53 | \( 1 - 2.69e8iT - 1.74e17T^{2} \) |
| 59 | \( 1 - 1.44e8iT - 5.11e17T^{2} \) |
| 61 | \( 1 + 7.44e8T + 7.13e17T^{2} \) |
| 67 | \( 1 + 2.00e9T + 1.82e18T^{2} \) |
| 71 | \( 1 - 1.82e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 - 4.08e8T + 4.29e18T^{2} \) |
| 79 | \( 1 - 4.48e9T + 9.46e18T^{2} \) |
| 83 | \( 1 + 1.61e9iT - 1.55e19T^{2} \) |
| 89 | \( 1 - 6.49e9iT - 3.11e19T^{2} \) |
| 97 | \( 1 + 4.67e9T + 7.37e19T^{2} \) |
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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
−16.00100842213400193246320448350, −14.74207755492736138214948958525, −14.04598078806468442628335771986, −12.33769915997902721707352303025, −10.63668284906299072693354036954, −9.076463950094602349348404946392, −7.46207444720723752078504697844, −6.58762726328646699001188327682, −5.13420132046770721805652738162, −2.86391836810384418926439977842,
0.13746641688915828099278987639, 1.76915548266894619512402468773, 3.32535002357816767187667219015, 4.99406149139287821271960488010, 7.24619258464077811735913734598, 9.359213106904556318143531900004, 10.04115656653760860810103655221, 11.74012928197011211210030214844, 12.48290463082344447922377006945, 13.53660601404652179047481374819
