L(s) = 1 | + (−0.144 + 1.40i)2-s − 3-s + (−1.95 − 0.406i)4-s + (0.144 − 1.40i)6-s − 3.62i·7-s + (0.855 − 2.69i)8-s + 9-s + 6.20i·11-s + (1.95 + 0.406i)12-s − 0.578·13-s + (5.10 + 0.524i)14-s + (3.66 + 1.59i)16-s + 1.42i·17-s + (−0.144 + 1.40i)18-s + 5.62i·19-s + ⋯ |
L(s) = 1 | + (−0.102 + 0.994i)2-s − 0.577·3-s + (−0.979 − 0.203i)4-s + (0.0590 − 0.574i)6-s − 1.37i·7-s + (0.302 − 0.953i)8-s + 0.333·9-s + 1.87i·11-s + (0.565 + 0.117i)12-s − 0.160·13-s + (1.36 + 0.140i)14-s + (0.917 + 0.398i)16-s + 0.344i·17-s + (−0.0340 + 0.331i)18-s + 1.29i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.717 - 0.696i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.717 - 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.300045 + 0.739497i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.300045 + 0.739497i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.144 - 1.40i)T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 3.62iT - 7T^{2} \) |
| 11 | \( 1 - 6.20iT - 11T^{2} \) |
| 13 | \( 1 + 0.578T + 13T^{2} \) |
| 17 | \( 1 - 1.42iT - 17T^{2} \) |
| 19 | \( 1 - 5.62iT - 19T^{2} \) |
| 23 | \( 1 - 5.62iT - 23T^{2} \) |
| 29 | \( 1 + 2iT - 29T^{2} \) |
| 31 | \( 1 + 2.57T + 31T^{2} \) |
| 37 | \( 1 - 7.83T + 37T^{2} \) |
| 41 | \( 1 - 5.25T + 41T^{2} \) |
| 43 | \( 1 + 7.25T + 43T^{2} \) |
| 47 | \( 1 - 6.78iT - 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 - 2.20iT - 59T^{2} \) |
| 61 | \( 1 - 12.4iT - 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 8.41T + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 + 5.42T + 79T^{2} \) |
| 83 | \( 1 - 3.25T + 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 - 4.84iT - 97T^{2} \) |
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
−10.68478174294418379747816060220, −9.967124015259165221254840775192, −9.473566897626017550280921785378, −7.85476970944891725221182309500, −7.45922572358450266777781150226, −6.64609790711730922408455178439, −5.62127773430141346814980798220, −4.51384057681432099411117707924, −3.93153066843160318929114544702, −1.41435310844905832860546651901,
0.54388433427110706472065004062, 2.37782268559277801117909772703, 3.29705382881241156709347973515, 4.77537189873276606493055123896, 5.55728717117692810265656600414, 6.46401306264438788070095442518, 8.070886150020945909963537640349, 8.836255929669702365732464080210, 9.409860630105957912173617213996, 10.64675645213611373785847268377