| L(s) = 1 | − 50.9i·2-s + 197. i·3-s − 1.57e3·4-s − 5.53e3i·5-s + 1.00e4·6-s + 2.13e4i·7-s + 2.80e4i·8-s + 1.99e4·9-s − 2.81e5·10-s − 1.31e5·11-s − 3.11e5i·12-s − 5.88e5·13-s + 1.09e6·14-s + 1.09e6·15-s − 1.81e5·16-s − 2.70e5·17-s + ⋯ |
| L(s) = 1 | − 1.59i·2-s + 0.814i·3-s − 1.53·4-s − 1.76i·5-s + 1.29·6-s + 1.27i·7-s + 0.857i·8-s + 0.337·9-s − 2.81·10-s − 0.813·11-s − 1.25i·12-s − 1.58·13-s + 2.02·14-s + 1.44·15-s − 0.172·16-s − 0.190·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 - 0.533i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.846 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.581972 + 0.168032i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.581972 + 0.168032i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (1.24e8 - 7.83e7i)T \) |
| good | 2 | \( 1 + 50.9iT - 1.02e3T^{2} \) |
| 3 | \( 1 - 197. iT - 5.90e4T^{2} \) |
| 5 | \( 1 + 5.53e3iT - 9.76e6T^{2} \) |
| 7 | \( 1 - 2.13e4iT - 2.82e8T^{2} \) |
| 11 | \( 1 + 1.31e5T + 2.59e10T^{2} \) |
| 13 | \( 1 + 5.88e5T + 1.37e11T^{2} \) |
| 17 | \( 1 + 2.70e5T + 2.01e12T^{2} \) |
| 19 | \( 1 - 2.93e6iT - 6.13e12T^{2} \) |
| 23 | \( 1 - 1.08e7T + 4.14e13T^{2} \) |
| 29 | \( 1 + 7.43e6iT - 4.20e14T^{2} \) |
| 31 | \( 1 - 4.62e7T + 8.19e14T^{2} \) |
| 37 | \( 1 - 6.79e7iT - 4.80e15T^{2} \) |
| 41 | \( 1 - 7.10e7T + 1.34e16T^{2} \) |
| 47 | \( 1 + 3.33e8T + 5.25e16T^{2} \) |
| 53 | \( 1 + 3.32e8T + 1.74e17T^{2} \) |
| 59 | \( 1 + 7.48e7T + 5.11e17T^{2} \) |
| 61 | \( 1 - 7.63e8iT - 7.13e17T^{2} \) |
| 67 | \( 1 - 5.79e8T + 1.82e18T^{2} \) |
| 71 | \( 1 - 9.90e8iT - 3.25e18T^{2} \) |
| 73 | \( 1 + 8.61e8iT - 4.29e18T^{2} \) |
| 79 | \( 1 + 4.77e9T + 9.46e18T^{2} \) |
| 83 | \( 1 + 1.89e9T + 1.55e19T^{2} \) |
| 89 | \( 1 - 1.53e9iT - 3.11e19T^{2} \) |
| 97 | \( 1 + 1.02e10T + 7.37e19T^{2} \) |
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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
−13.09980956694168452395320528919, −12.50177130459597587401822905072, −11.69434757943850099401892282969, −10.02944870303087632037833833756, −9.455133204143852402268452304012, −8.365167381543725122224010116894, −5.12692756723572785191655558831, −4.57740032168214080314792014591, −2.76538519659414784040187339850, −1.37764731546882843352867658393,
0.20026816111082491605668673573, 2.67643417536565340648204979096, 4.77247777656190958486061492289, 6.71789183884632889947821233085, 7.06928887488424587104169870168, 7.75026857705843988477712281575, 9.920389214639071593147379646586, 11.07491384987218884286657457593, 13.09419108410607017300997040676, 13.95114490380880224851264434384
