| L(s) = 1 | − 2.82·2-s + 8.00·4-s − 83i·7-s − 22.6·8-s + 80.6i·11-s + 41i·13-s + 234. i·14-s + 64.0·16-s + 513.·17-s + 139·19-s − 228i·22-s + 224.·23-s − 115. i·26-s − 664. i·28-s − 674. i·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.500·4-s − 1.69i·7-s − 0.353·8-s + 0.666i·11-s + 0.242i·13-s + 1.19i·14-s + 0.250·16-s + 1.77·17-s + 0.385·19-s − 0.471i·22-s + 0.425·23-s − 0.171i·26-s − 0.846i·28-s − 0.802i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.151 + 0.988i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.151 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.366890581\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.366890581\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2.82T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 83iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 80.6iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 41iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 513.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 139T + 1.30e5T^{2} \) |
| 23 | \( 1 - 224.T + 2.79e5T^{2} \) |
| 29 | \( 1 + 674. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 1.05e3T + 9.23e5T^{2} \) |
| 37 | \( 1 - 1.67e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 831. iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 2.51e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 2.94e3T + 4.87e6T^{2} \) |
| 53 | \( 1 - 390.T + 7.89e6T^{2} \) |
| 59 | \( 1 + 750. iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 5.82e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 7.25e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 6.90e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 4.55e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 9.29e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 7.98e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + 6.07e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.79e3iT - 8.85e7T^{2} \) |
| 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
−10.06646440294401951825352652683, −9.709312096827111192188785059697, −8.350961590651176272440431740196, −7.39409357648489406416420661491, −7.03209318819125423244138957766, −5.61341860390395767981186567968, −4.29559554103708375803458128876, −3.25879451802928081621423740044, −1.56583841427595714268239823217, −0.56363721924432335366667695517,
1.08314282306838119594467977647, 2.47585866711124086027126656225, 3.42980527757716331507127363669, 5.46604557303699875696196799300, 5.74668803600487791565020845764, 7.17534309016769566013189804598, 8.145166648264497327235196204114, 8.921035904475261201467673259815, 9.568331171404757926432540715638, 10.63087349854173889812543448137
