| L(s) = 1 | + 2.69i·2-s − 5.25·4-s + (1.80 + 1.32i)5-s − 4.61i·7-s − 8.76i·8-s + (−3.56 + 4.85i)10-s + 1.48·11-s − 6.10i·13-s + 12.4·14-s + 13.1·16-s − i·17-s − 0.180·19-s + (−9.47 − 6.94i)20-s + 4.00i·22-s − 6.74i·23-s + ⋯ |
| L(s) = 1 | + 1.90i·2-s − 2.62·4-s + (0.806 + 0.591i)5-s − 1.74i·7-s − 3.09i·8-s + (−1.12 + 1.53i)10-s + 0.448·11-s − 1.69i·13-s + 3.32·14-s + 3.27·16-s − 0.242i·17-s − 0.0415·19-s + (−2.11 − 1.55i)20-s + 0.854i·22-s − 1.40i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 765 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.591 - 0.806i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 765 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.591 - 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.18132 + 0.598648i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.18132 + 0.598648i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-1.80 - 1.32i)T \) |
| 17 | \( 1 + iT \) |
| good | 2 | \( 1 - 2.69iT - 2T^{2} \) |
| 7 | \( 1 + 4.61iT - 7T^{2} \) |
| 11 | \( 1 - 1.48T + 11T^{2} \) |
| 13 | \( 1 + 6.10iT - 13T^{2} \) |
| 19 | \( 1 + 0.180T + 19T^{2} \) |
| 23 | \( 1 + 6.74iT - 23T^{2} \) |
| 29 | \( 1 + 1.31T + 29T^{2} \) |
| 31 | \( 1 - 3.02T + 31T^{2} \) |
| 37 | \( 1 + 2.11iT - 37T^{2} \) |
| 41 | \( 1 + 8.22T + 41T^{2} \) |
| 43 | \( 1 + 2.32iT - 43T^{2} \) |
| 47 | \( 1 - 6.56iT - 47T^{2} \) |
| 53 | \( 1 + 1.17iT - 53T^{2} \) |
| 59 | \( 1 - 3.42T + 59T^{2} \) |
| 61 | \( 1 + 10.1T + 61T^{2} \) |
| 67 | \( 1 - 12.6iT - 67T^{2} \) |
| 71 | \( 1 - 7.21T + 71T^{2} \) |
| 73 | \( 1 + 9.33iT - 73T^{2} \) |
| 79 | \( 1 - 8.76T + 79T^{2} \) |
| 83 | \( 1 - 13.0iT - 83T^{2} \) |
| 89 | \( 1 - 8.43T + 89T^{2} \) |
| 97 | \( 1 - 6.28iT - 97T^{2} \) |
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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
−10.25335886816820839267738666633, −9.508556404676851026124398870996, −8.374083336956610446457430094251, −7.64409338518327163233654231091, −6.90210860980453602408659808636, −6.32647323896294065100409843386, −5.35504860460879709090384322526, −4.41215188161452783057266401149, −3.33081291823303192032454792359, −0.72394293938527811490143042136,
1.64152638825099713900459237845, 2.12946606444136448748803758309, 3.37963085762741814524669807842, 4.60295478821544475441452398248, 5.35549578656471987946338809041, 6.34021008511847801682498150014, 8.311838601351532504001897134387, 9.111529979598397992288187804646, 9.299607811958025210604023444136, 10.10351846195345767446048494081
