| L(s) = 1 | − 5.65i·2-s − 32.0·4-s + 233. i·5-s − 191.·7-s + 181. i·8-s + 1.32e3·10-s + 777. i·11-s − 91.1·13-s + 1.08e3i·14-s + 1.02e3·16-s + 7.04e3i·17-s + 2.73e3·19-s − 7.47e3i·20-s + 4.39e3·22-s − 1.98e4i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s + 1.86i·5-s − 0.557·7-s + 0.353i·8-s + 1.32·10-s + 0.583i·11-s − 0.0414·13-s + 0.394i·14-s + 0.250·16-s + 1.43i·17-s + 0.398·19-s − 0.934i·20-s + 0.412·22-s − 1.63i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.3115894694\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3115894694\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 5.65iT \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 233. iT - 1.56e4T^{2} \) |
| 7 | \( 1 + 191.T + 1.17e5T^{2} \) |
| 11 | \( 1 - 777. iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 91.1T + 4.82e6T^{2} \) |
| 17 | \( 1 - 7.04e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 2.73e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + 1.98e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 3.12e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 1.23e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + 2.79e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + 4.32e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 3.85e4T + 6.32e9T^{2} \) |
| 47 | \( 1 - 1.66e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 5.47e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 1.62e4iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 5.88e4T + 5.15e10T^{2} \) |
| 67 | \( 1 - 2.95e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 1.57e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 8.02e4T + 1.51e11T^{2} \) |
| 79 | \( 1 + 3.76e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + 8.47e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 1.12e3iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 1.35e6T + 8.32e11T^{2} \) |
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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
−12.15280276794188666362822704885, −11.03462848351320702945832002042, −10.38147305341091776034085502278, −9.663325059262228965142950341726, −8.111723622743740809089593813109, −6.89473943929189273382110063818, −6.04147356852022015962643179697, −4.12371358920006808958781143869, −3.06459662718553431769336398445, −2.05774672928363135201491224294,
0.095475165439107824285379412825, 1.25117904604688978779401430077, 3.52515852531809923106811634174, 4.98436901634574371823439060157, 5.55309350504944183307386085361, 7.07758912343395078448006021284, 8.215875779397771477161385845494, 9.136071937024225902267693448569, 9.700198254312030944013260535864, 11.51290048595463973759303986678
