| L(s) = 1 | + 2.75i·3-s − 3.57i·7-s − 4.57·9-s − 11-s − i·13-s + 0.751i·17-s − 2.50·19-s + 9.82·21-s + 5.75i·23-s − 4.32i·27-s + 4.07·29-s + 6.14·31-s − 2.75i·33-s − 2.81i·37-s + 2.75·39-s + ⋯ |
| L(s) = 1 | + 1.58i·3-s − 1.34i·7-s − 1.52·9-s − 0.301·11-s − 0.277i·13-s + 0.182i·17-s − 0.574·19-s + 2.14·21-s + 1.19i·23-s − 0.831i·27-s + 0.756·29-s + 1.10·31-s − 0.478i·33-s − 0.463i·37-s + 0.440·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.165790560\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.165790560\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 2.75iT - 3T^{2} \) |
| 7 | \( 1 + 3.57iT - 7T^{2} \) |
| 13 | \( 1 + iT - 13T^{2} \) |
| 17 | \( 1 - 0.751iT - 17T^{2} \) |
| 19 | \( 1 + 2.50T + 19T^{2} \) |
| 23 | \( 1 - 5.75iT - 23T^{2} \) |
| 29 | \( 1 - 4.07T + 29T^{2} \) |
| 31 | \( 1 - 6.14T + 31T^{2} \) |
| 37 | \( 1 + 2.81iT - 37T^{2} \) |
| 41 | \( 1 - 1.18T + 41T^{2} \) |
| 43 | \( 1 - 7.68iT - 43T^{2} \) |
| 47 | \( 1 - 9.82iT - 47T^{2} \) |
| 53 | \( 1 + 14.2iT - 53T^{2} \) |
| 59 | \( 1 + 9.82T + 59T^{2} \) |
| 61 | \( 1 - 7.07T + 61T^{2} \) |
| 67 | \( 1 - 14.6iT - 67T^{2} \) |
| 71 | \( 1 + 5.81T + 71T^{2} \) |
| 73 | \( 1 - 14.7iT - 73T^{2} \) |
| 79 | \( 1 + 3.89T + 79T^{2} \) |
| 83 | \( 1 + 5.57iT - 83T^{2} \) |
| 89 | \( 1 + 7.42T + 89T^{2} \) |
| 97 | \( 1 + 0.609iT - 97T^{2} \) |
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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
−8.716947818464635202745823706526, −8.054984900059332959201133780917, −7.30440566358542473117816087040, −6.39748594391727241925909291457, −5.53794874085496161350821427112, −4.71261472962224850790713870824, −4.20876767843263932215344618773, −3.54117965806974895970938042529, −2.72023893005642295635903899752, −1.12308700942583563045841123194,
0.34877819004811475506153239958, 1.62421498171095563732141284565, 2.42783282516369637136825252636, 2.92117273973427781767713476079, 4.41238173776990862248214274799, 5.29095962517749090698983884720, 6.15486631022406990608669673282, 6.48947865246488995760373419142, 7.28971493020779253349282350611, 8.115628131014498206874747421797
