L(s) = 1 | + 4.53i·2-s − 3i·3-s − 12.5·4-s + 13.5·6-s − 7i·7-s − 20.5i·8-s − 9·9-s − 19.0·11-s + 37.5i·12-s + 2.93i·13-s + 31.7·14-s − 7.21·16-s − 6.49i·17-s − 40.7i·18-s + 5.43·19-s + ⋯ |
L(s) = 1 | + 1.60i·2-s − 0.577i·3-s − 1.56·4-s + 0.924·6-s − 0.377i·7-s − 0.907i·8-s − 0.333·9-s − 0.522·11-s + 0.904i·12-s + 0.0626i·13-s + 0.605·14-s − 0.112·16-s − 0.0927i·17-s − 0.533i·18-s + 0.0656·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.595828317\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.595828317\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 3iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 7iT \) |
good | 2 | \( 1 - 4.53iT - 8T^{2} \) |
| 11 | \( 1 + 19.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 2.93iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 6.49iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 5.43T + 6.85e3T^{2} \) |
| 23 | \( 1 + 49.3iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 291.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 244.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 193. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 315.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 300. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 86.5iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 509. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 83.3T + 2.05e5T^{2} \) |
| 61 | \( 1 + 5.25T + 2.26e5T^{2} \) |
| 67 | \( 1 - 205. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 1.00e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.00e3iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 863.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.33e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 326.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.52e3iT - 9.12e5T^{2} \) |
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
−10.45329260520514514942915328639, −9.401933122833128782539599914653, −8.301017886816017981260461214780, −7.891179616727495944599954178332, −6.85758866460490217686195210065, −6.31714700248074349796356714708, −5.24789212667538642918090425330, −4.33430164799673217844166853276, −2.62292098925399939831972819075, −0.69665451496178968429292956999,
0.868343270078126560103328904762, 2.37717022641922166950124653938, 3.16997452289445418459853478361, 4.31358716731988401315353683963, 5.15095990825133038978404529576, 6.45658335329951133032491410051, 8.016587974664946570600668043427, 8.887331437523068612966163298906, 9.757080419167479103519571713321, 10.37886639460685881227761049474