| L(s) = 1 | + 8.13·5-s − 13.7·7-s − 214.·11-s + 116. i·13-s + 396.·17-s + (−24.2 + 360. i)19-s + 541.·23-s − 558.·25-s − 249. i·29-s − 416. i·31-s − 111.·35-s − 2.60e3i·37-s − 1.04e3i·41-s − 1.73e3·43-s + 3.60e3·47-s + ⋯ |
| L(s) = 1 | + 0.325·5-s − 0.280·7-s − 1.77·11-s + 0.688i·13-s + 1.37·17-s + (−0.0672 + 0.997i)19-s + 1.02·23-s − 0.894·25-s − 0.296i·29-s − 0.433i·31-s − 0.0911·35-s − 1.90i·37-s − 0.621i·41-s − 0.937·43-s + 1.63·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0672 + 0.997i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.0672 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.102060065\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.102060065\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (24.2 - 360. i)T \) |
| good | 5 | \( 1 - 8.13T + 625T^{2} \) |
| 7 | \( 1 + 13.7T + 2.40e3T^{2} \) |
| 11 | \( 1 + 214.T + 1.46e4T^{2} \) |
| 13 | \( 1 - 116. iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 396.T + 8.35e4T^{2} \) |
| 23 | \( 1 - 541.T + 2.79e5T^{2} \) |
| 29 | \( 1 + 249. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 416. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 2.60e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + 1.04e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 1.73e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 3.60e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + 4.22e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 6.18e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 3.14e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 5.59e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 7.56e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 6.23e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 2.16e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 2.81e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + 1.29e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.11e4iT - 8.85e7T^{2} \) |
| 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
−9.892303364294246650634459000119, −8.817195890843990648876608169083, −7.82735370701663231504609943426, −7.20323925369116476467868143113, −5.82079976785144317628550475283, −5.38506920027305893568762077027, −4.03240863101793752337247411778, −2.90857072775330610190516909296, −1.83415558366118937689698170042, −0.28841910004745412682572968651,
1.02406453563650551044483383932, 2.60194288430202629860472174798, 3.28178127163461864150257426360, 4.95002947362846297519002256406, 5.43817634820933298501981438065, 6.57232110286308073177209636548, 7.65048847668392231004767422853, 8.224237121243389803522790111798, 9.404369460828923271936205714053, 10.18540423753824980956160173759
