L(s) = 1 | − 0.221i·3-s + 22.7i·7-s + 26.9·9-s − 66.8·11-s − 32.9i·13-s − 35.9i·17-s − 44.0·19-s + 5.04·21-s − 139. i·23-s − 11.9i·27-s − 134.·29-s + 229.·31-s + 14.8i·33-s − 79.1i·37-s − 7.29·39-s + ⋯ |
L(s) = 1 | − 0.0426i·3-s + 1.23i·7-s + 0.998·9-s − 1.83·11-s − 0.702i·13-s − 0.512i·17-s − 0.531·19-s + 0.0524·21-s − 1.26i·23-s − 0.0851i·27-s − 0.863·29-s + 1.32·31-s + 0.0780i·33-s − 0.351i·37-s − 0.0299·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.426817362\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.426817362\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + 0.221iT - 27T^{2} \) |
| 7 | \( 1 - 22.7iT - 343T^{2} \) |
| 11 | \( 1 + 66.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 32.9iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 35.9iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 44.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 139. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 134.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 229.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 79.1iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 384.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 251. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 241. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 222iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 552.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 494.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 574. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 654.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.13e3iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 179.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 810. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 29.7T + 7.04e5T^{2} \) |
| 97 | \( 1 + 383. iT - 9.12e5T^{2} \) |
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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
−9.894505873900044228885597909878, −8.766475386179332999142859518862, −8.061858431639378674748567537297, −7.25318194924340588398979364805, −6.08325499442506810631025478255, −5.28792507681093359262842104045, −4.44506432147074495976642057399, −2.87020620664953497978229698944, −2.20846302737937141140596658127, −0.43075505194852935755442501143,
1.04268803124050707868490558317, 2.33361115477390013944692328700, 3.77394570633646915210492292276, 4.48533733723968547052454447367, 5.52633480280075692441558228842, 6.74054109274776608153950453279, 7.51020270316209379731215046864, 8.055023926537961737190333327147, 9.357442898338549991195352650742, 10.22309788103046010290100547408