| L(s) = 1 | + (−5.00 − 1.40i)3-s + 5.86i·5-s − 5.92i·7-s + (23.0 + 14.0i)9-s − 27.9·11-s + 0.0653·13-s + (8.26 − 29.3i)15-s + 36.9i·17-s + 30.7i·19-s + (−8.33 + 29.6i)21-s + 61.2·23-s + 90.5·25-s + (−95.3 − 102. i)27-s + 143. i·29-s − 299. i·31-s + ⋯ |
| L(s) = 1 | + (−0.962 − 0.270i)3-s + 0.524i·5-s − 0.319i·7-s + (0.853 + 0.521i)9-s − 0.766·11-s + 0.00139·13-s + (0.142 − 0.505i)15-s + 0.527i·17-s + 0.371i·19-s + (−0.0866 + 0.307i)21-s + 0.555·23-s + 0.724·25-s + (−0.679 − 0.733i)27-s + 0.919i·29-s − 1.73i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.270 + 0.962i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.270 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6929221525\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6929221525\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (5.00 + 1.40i)T \) |
| good | 5 | \( 1 - 5.86iT - 125T^{2} \) |
| 7 | \( 1 + 5.92iT - 343T^{2} \) |
| 11 | \( 1 + 27.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 0.0653T + 2.19e3T^{2} \) |
| 17 | \( 1 - 36.9iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 30.7iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 61.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 143. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 299. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 340.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 379. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 470. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 428.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 505. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 207.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 578.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 415. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 547.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 194.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 308. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 62.3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.06e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 703.T + 9.12e5T^{2} \) |
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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
−10.58038968725572770736292051097, −10.21577219425638586297699635154, −8.752217429482683792855154324538, −7.53558428627786318439845538522, −6.88579616654700776031318464764, −5.81541407803659936935922142367, −4.91481340032966530173421948067, −3.56120166615933291651562507386, −1.94692426535217421073659798049, −0.30126268005233206738770217167,
1.16564884307451465270989802327, 3.00314582814298782128940102099, 4.65536295259646488682838777768, 5.17572901045743789533130434700, 6.30430124546010955450248175270, 7.30139114317532513513468668022, 8.523577915854538063753177560824, 9.449211585476402003949792597647, 10.38237914792899078796534978481, 11.17278930108646271394838795210
