| L(s) = 1 | + (0.615 + 1.61i)3-s − 1.88i·5-s − 0.545i·7-s + (−2.24 + 1.99i)9-s − 2.45·11-s + 5.44·13-s + (3.05 − 1.16i)15-s − 5.91i·17-s − i·19-s + (0.882 − 0.335i)21-s − 1.00·23-s + 1.43·25-s + (−4.60 − 2.40i)27-s − 0.590i·29-s + 1.22i·31-s + ⋯ |
| L(s) = 1 | + (0.355 + 0.934i)3-s − 0.844i·5-s − 0.206i·7-s + (−0.747 + 0.664i)9-s − 0.741·11-s + 1.51·13-s + (0.789 − 0.300i)15-s − 1.43i·17-s − 0.229i·19-s + (0.192 − 0.0732i)21-s − 0.208·23-s + 0.286·25-s + (−0.886 − 0.461i)27-s − 0.109i·29-s + 0.219i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1824 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 + 0.409i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1824 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.912 + 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.828859270\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.828859270\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.615 - 1.61i)T \) |
| 19 | \( 1 + iT \) |
| good | 5 | \( 1 + 1.88iT - 5T^{2} \) |
| 7 | \( 1 + 0.545iT - 7T^{2} \) |
| 11 | \( 1 + 2.45T + 11T^{2} \) |
| 13 | \( 1 - 5.44T + 13T^{2} \) |
| 17 | \( 1 + 5.91iT - 17T^{2} \) |
| 23 | \( 1 + 1.00T + 23T^{2} \) |
| 29 | \( 1 + 0.590iT - 29T^{2} \) |
| 31 | \( 1 - 1.22iT - 31T^{2} \) |
| 37 | \( 1 - 2.77T + 37T^{2} \) |
| 41 | \( 1 + 5.58iT - 41T^{2} \) |
| 43 | \( 1 + 2.46iT - 43T^{2} \) |
| 47 | \( 1 - 4.49T + 47T^{2} \) |
| 53 | \( 1 + 8.61iT - 53T^{2} \) |
| 59 | \( 1 - 5.63T + 59T^{2} \) |
| 61 | \( 1 - 6.67T + 61T^{2} \) |
| 67 | \( 1 + 11.7iT - 67T^{2} \) |
| 71 | \( 1 + 1.56T + 71T^{2} \) |
| 73 | \( 1 + 5.18T + 73T^{2} \) |
| 79 | \( 1 - 14.3iT - 79T^{2} \) |
| 83 | \( 1 - 14.1T + 83T^{2} \) |
| 89 | \( 1 - 1.32iT - 89T^{2} \) |
| 97 | \( 1 + 17.3T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.114862410822065979606298531873, −8.597241523352464018099896120080, −7.910507938937068674249769460350, −6.88444422026623054046182353104, −5.64665570583905312631871580032, −5.11161573646161042520485145815, −4.25115319885610644635882176364, −3.41427078247445751218264179355, −2.35052843994091520538199050766, −0.72535606154613585052692657735,
1.23300390696522415966436353279, 2.36058753509324445135492416772, 3.22273055569116869091542255653, 4.09148729855834775706289857465, 5.70664268198244651829995408011, 6.14396399792652878412876019643, 6.92577927494454737330039939493, 7.81522478361602429930575934701, 8.379555484739200180203490462669, 9.057742955477361223400111858726
