| L(s) = 1 | + 1.41·2-s + 2.00·4-s − 2.23i·5-s − 8.51i·7-s + 2.82·8-s − 3.16i·10-s − 7.57i·11-s − 2.64·13-s − 12.0i·14-s + 4.00·16-s + 7.56i·17-s − 24.2i·19-s − 4.47i·20-s − 10.7i·22-s + (15.7 + 16.7i)23-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.500·4-s − 0.447i·5-s − 1.21i·7-s + 0.353·8-s − 0.316i·10-s − 0.688i·11-s − 0.203·13-s − 0.859i·14-s + 0.250·16-s + 0.444i·17-s − 1.27i·19-s − 0.223i·20-s − 0.486i·22-s + (0.684 + 0.729i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.729 + 0.684i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.729 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.419781687\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.419781687\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 23 | \( 1 + (-15.7 - 16.7i)T \) |
| good | 7 | \( 1 + 8.51iT - 49T^{2} \) |
| 11 | \( 1 + 7.57iT - 121T^{2} \) |
| 13 | \( 1 + 2.64T + 169T^{2} \) |
| 17 | \( 1 - 7.56iT - 289T^{2} \) |
| 19 | \( 1 + 24.2iT - 361T^{2} \) |
| 29 | \( 1 - 31.8T + 841T^{2} \) |
| 31 | \( 1 + 56.5T + 961T^{2} \) |
| 37 | \( 1 + 39.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 42.5T + 1.68e3T^{2} \) |
| 43 | \( 1 + 20.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 84.3T + 2.20e3T^{2} \) |
| 53 | \( 1 - 11.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 67.6T + 3.48e3T^{2} \) |
| 61 | \( 1 - 35.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 44.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 8.86T + 5.04e3T^{2} \) |
| 73 | \( 1 + 87.4T + 5.32e3T^{2} \) |
| 79 | \( 1 - 154. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 141. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 63.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 143. iT - 9.40e3T^{2} \) |
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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
−8.668066683454638570019691424237, −7.58668314442457328623110488579, −7.14183567679617873022729855854, −6.20132778214896493792823604537, −5.31919352389082213571897048856, −4.54561236668460231218309895639, −3.77571108766823225129791981098, −2.91881650249419369086527265757, −1.53882265571844055228195803757, −0.44401185053568768921721253735,
1.64166958677352467335897095963, 2.57887688145409560725792801183, 3.31748161241545314381679395141, 4.49183697083173849664417223948, 5.21637910375183028145964599381, 6.03633295083301334203230745272, 6.70818198390998913013938706155, 7.58228450553028685594512609167, 8.377033277307508312785492409400, 9.302040260076686090751181898870
