| L(s) = 1 | + 42.0i·2-s − 743.·4-s + 4.85e3i·5-s + 3.18e4·7-s + 1.17e4i·8-s − 2.04e5·10-s + 1.14e5i·11-s + 2.66e5·13-s + 1.33e6i·14-s − 1.25e6·16-s − 1.95e5i·17-s − 1.34e6·19-s − 3.60e6i·20-s − 4.82e6·22-s − 9.10e6i·23-s + ⋯ |
| L(s) = 1 | + 1.31i·2-s − 0.726·4-s + 1.55i·5-s + 1.89·7-s + 0.359i·8-s − 2.04·10-s + 0.712i·11-s + 0.718·13-s + 2.48i·14-s − 1.19·16-s − 0.137i·17-s − 0.543·19-s − 1.12i·20-s − 0.936·22-s − 1.41i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(2.25683i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.25683i\) |
| \(L(6)\) |
|
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 | \( 1 - 42.0iT - 1.02e3T^{2} \) |
| 5 | \( 1 - 4.85e3iT - 9.76e6T^{2} \) |
| 7 | \( 1 - 3.18e4T + 2.82e8T^{2} \) |
| 11 | \( 1 - 1.14e5iT - 2.59e10T^{2} \) |
| 13 | \( 1 - 2.66e5T + 1.37e11T^{2} \) |
| 17 | \( 1 + 1.95e5iT - 2.01e12T^{2} \) |
| 19 | \( 1 + 1.34e6T + 6.13e12T^{2} \) |
| 23 | \( 1 + 9.10e6iT - 4.14e13T^{2} \) |
| 29 | \( 1 + 9.95e6iT - 4.20e14T^{2} \) |
| 31 | \( 1 + 5.35e6T + 8.19e14T^{2} \) |
| 37 | \( 1 - 4.26e7T + 4.80e15T^{2} \) |
| 41 | \( 1 - 1.71e8iT - 1.34e16T^{2} \) |
| 43 | \( 1 + 5.96e7T + 2.16e16T^{2} \) |
| 47 | \( 1 + 3.45e8iT - 5.25e16T^{2} \) |
| 53 | \( 1 - 3.02e8iT - 1.74e17T^{2} \) |
| 59 | \( 1 - 4.87e8iT - 5.11e17T^{2} \) |
| 61 | \( 1 + 1.61e8T + 7.13e17T^{2} \) |
| 67 | \( 1 + 4.59e8T + 1.82e18T^{2} \) |
| 71 | \( 1 + 2.12e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 - 3.50e9T + 4.29e18T^{2} \) |
| 79 | \( 1 + 2.47e8T + 9.46e18T^{2} \) |
| 83 | \( 1 + 5.42e9iT - 1.55e19T^{2} \) |
| 89 | \( 1 + 8.33e8iT - 3.11e19T^{2} \) |
| 97 | \( 1 + 5.68e9T + 7.37e19T^{2} \) |
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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
−15.12062330711458273576398553608, −14.82206845733968832220320073069, −13.88907889810931922695537637167, −11.52159255813395780380371464013, −10.62634318770204970037297968536, −8.386058543554637093984106637832, −7.39627252186228588368512297815, −6.25465216047018460901184955974, −4.61756000643876926657317915625, −2.18098871992400580126171673448,
0.950574155844347118610784672748, 1.74599005879103575891419110916, 4.02586383465118008874193024882, 5.28210486907589864207539502830, 8.125252587231725188713924652109, 9.104962043647513056104230114362, 10.91137634972432910554257574568, 11.66134651185696473755463022665, 12.82198089573100851913757011762, 13.93464946231908262844665978800
