| L(s) = 1 | − 3.44i·2-s + 4.13·4-s + 16.2·5-s + 92.6·7-s − 69.3i·8-s − 55.8i·10-s + 48.6i·11-s − 216. i·13-s − 319. i·14-s − 172.·16-s − 171.·17-s + 267.·19-s + 67.1·20-s + 167.·22-s − 734. i·23-s + ⋯ |
| L(s) = 1 | − 0.861i·2-s + 0.258·4-s + 0.648·5-s + 1.89·7-s − 1.08i·8-s − 0.558i·10-s + 0.402i·11-s − 1.28i·13-s − 1.62i·14-s − 0.674·16-s − 0.595·17-s + 0.740·19-s + 0.167·20-s + 0.346·22-s − 1.38i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.380 + 0.924i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.380 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(3.489209720\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.489209720\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 + (-1.32e3 + 3.21e3i)T \) |
| good | 2 | \( 1 + 3.44iT - 16T^{2} \) |
| 5 | \( 1 - 16.2T + 625T^{2} \) |
| 7 | \( 1 - 92.6T + 2.40e3T^{2} \) |
| 11 | \( 1 - 48.6iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 216. iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 171.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 267.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 734. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 999.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 568. iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 1.00e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 2.30e3T + 2.82e6T^{2} \) |
| 43 | \( 1 - 2.15e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 1.19e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 1.10e3T + 7.89e6T^{2} \) |
| 61 | \( 1 - 4.03e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 8.14e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 2.53e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 792. iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 3.21e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 1.05e4iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 226. iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 7.28e3iT - 8.85e7T^{2} \) |
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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
−10.20333412550284331268941585900, −9.333169687115794266050627644320, −8.103664384976849100058249563034, −7.47306937944569715885567530633, −6.14789945576178821434983772029, −5.14367457458397358127370670524, −4.15585399921826209793011374346, −2.65614198933273618515107589392, −1.87690960279627635128427612282, −0.878728081671032775139294967070,
1.54615409831733517586044600741, 2.18719101012990617658983144286, 4.09746121382433719345810568930, 5.29858011149401130955724839234, 5.75109945302540882998693409703, 7.07122069806491633131062146205, 7.62096897953862740964824278298, 8.616888110641147892104535316826, 9.329298824177644435672167602342, 10.72035739818541807279512360746
