| L(s) = 1 | − 30.3i·2-s − 212. i·3-s + 105.·4-s − 1.63e3i·5-s − 6.45e3·6-s + 7.42e3i·7-s − 3.42e4i·8-s + 1.37e4·9-s − 4.94e4·10-s − 1.29e5·11-s − 2.24e4i·12-s − 4.77e5·13-s + 2.25e5·14-s − 3.47e5·15-s − 9.29e5·16-s − 2.39e5·17-s + ⋯ |
| L(s) = 1 | − 0.947i·2-s − 0.876i·3-s + 0.102·4-s − 0.522i·5-s − 0.829·6-s + 0.441i·7-s − 1.04i·8-s + 0.232·9-s − 0.494·10-s − 0.805·11-s − 0.0900i·12-s − 1.28·13-s + 0.418·14-s − 0.457·15-s − 0.886·16-s − 0.168·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.406 - 0.913i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.406 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.618051 + 0.951193i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.618051 + 0.951193i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-5.97e7 - 1.34e8i)T \) |
| good | 2 | \( 1 + 30.3iT - 1.02e3T^{2} \) |
| 3 | \( 1 + 212. iT - 5.90e4T^{2} \) |
| 5 | \( 1 + 1.63e3iT - 9.76e6T^{2} \) |
| 7 | \( 1 - 7.42e3iT - 2.82e8T^{2} \) |
| 11 | \( 1 + 1.29e5T + 2.59e10T^{2} \) |
| 13 | \( 1 + 4.77e5T + 1.37e11T^{2} \) |
| 17 | \( 1 + 2.39e5T + 2.01e12T^{2} \) |
| 19 | \( 1 + 2.40e6iT - 6.13e12T^{2} \) |
| 23 | \( 1 + 1.07e7T + 4.14e13T^{2} \) |
| 29 | \( 1 - 3.05e7iT - 4.20e14T^{2} \) |
| 31 | \( 1 - 1.70e7T + 8.19e14T^{2} \) |
| 37 | \( 1 - 2.92e7iT - 4.80e15T^{2} \) |
| 41 | \( 1 + 6.33e7T + 1.34e16T^{2} \) |
| 47 | \( 1 - 1.73e7T + 5.25e16T^{2} \) |
| 53 | \( 1 + 9.26e7T + 1.74e17T^{2} \) |
| 59 | \( 1 + 8.79e8T + 5.11e17T^{2} \) |
| 61 | \( 1 + 1.35e9iT - 7.13e17T^{2} \) |
| 67 | \( 1 - 1.85e8T + 1.82e18T^{2} \) |
| 71 | \( 1 + 3.17e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 + 2.67e9iT - 4.29e18T^{2} \) |
| 79 | \( 1 - 5.12e9T + 9.46e18T^{2} \) |
| 83 | \( 1 + 9.86e8T + 1.55e19T^{2} \) |
| 89 | \( 1 - 5.62e8iT - 3.11e19T^{2} \) |
| 97 | \( 1 + 1.18e10T + 7.37e19T^{2} \) |
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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
−12.50064700194499239461911707944, −12.16488535718634817095411979117, −10.66921879555606778599625475896, −9.474909641900431455802763150281, −7.82691245926387876173416733327, −6.63416016844782745109634967300, −4.79014743511544297129830138604, −2.75249427574775830259825378147, −1.73099140754921972797733052857, −0.33414145984390626745334166683,
2.41465199832715270326488687033, 4.25082777761779501982091669728, 5.61137557331428127963407189711, 7.06211197029164993091564070736, 8.026864133352469682169227254411, 9.893495292470529137598865106832, 10.62579988747163589101341989771, 12.11727737667503141375899558984, 13.88486604985476951374611476868, 14.87153427586088526902501960468
