| L(s) = 1 | + (−3 − 26.8i)3-s − 160. i·5-s + 242·7-s + (−711. + 160. i)9-s + 1.77e3i·11-s + 2.61e3·13-s + (−4.32e3 + 482. i)15-s − 7.08e3i·17-s + 5.78e3·19-s + (−726 − 6.49e3i)21-s + 9.33e3i·23-s − 1.02e4·25-s + (6.45e3 + 1.85e4i)27-s + 1.23e4i·29-s − 2.04e4·31-s + ⋯ |
| L(s) = 1 | + (−0.111 − 0.993i)3-s − 1.28i·5-s + 0.705·7-s + (−0.975 + 0.220i)9-s + 1.33i·11-s + 1.19·13-s + (−1.28 + 0.143i)15-s − 1.44i·17-s + 0.843·19-s + (−0.0783 − 0.701i)21-s + 0.767i·23-s − 0.658·25-s + (0.327 + 0.944i)27-s + 0.508i·29-s − 0.686·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.111 + 0.993i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.111 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.991249 - 0.886600i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.991249 - 0.886600i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (3 + 26.8i)T \) |
| good | 5 | \( 1 + 160. iT - 1.56e4T^{2} \) |
| 7 | \( 1 - 242T + 1.17e5T^{2} \) |
| 11 | \( 1 - 1.77e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 2.61e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + 7.08e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 5.78e3T + 4.70e7T^{2} \) |
| 23 | \( 1 - 9.33e3iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 1.23e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 2.04e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + 4.67e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + 3.54e3iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 6.86e4T + 6.32e9T^{2} \) |
| 47 | \( 1 - 2.12e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 1.71e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 1.49e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 2.47e4T + 5.15e10T^{2} \) |
| 67 | \( 1 + 8.43e4T + 9.04e10T^{2} \) |
| 71 | \( 1 - 3.24e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 1.13e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 1.59e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 5.15e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 1.25e6iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 8.99e5T + 8.32e11T^{2} \) |
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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
−18.33138354074355335679510005186, −17.44279706571324928566279972491, −15.96993678323881762099418021014, −13.98102496691122809523182190508, −12.73078811539483569863722773241, −11.57086431172043709093243595173, −9.027896876429834557698923146066, −7.48211938871517979409192143359, −5.15974285034614258636696909309, −1.34556197032193776475925785718,
3.51724726098409809230850848348, 6.00561721160988747847490223028, 8.475158197285153385459826092224, 10.57250980380935038050991394375, 11.26904939890611636478438499806, 13.95915122558260520443534704384, 14.98072690933008424000978889044, 16.29788464568323149878833575002, 17.85413251129500264344186035738, 19.09678958311059878787706393003
