| L(s) = 1 | + 10.4·3-s + (17.9 − 52.9i)5-s + 86.7i·7-s − 134.·9-s − 258. i·11-s − 465.·13-s + (187. − 551. i)15-s + 2.02e3i·17-s − 1.05e3i·19-s + 904. i·21-s + 3.99e3i·23-s + (−2.47e3 − 1.90e3i)25-s − 3.93e3·27-s + 3.24e3i·29-s − 991.·31-s + ⋯ |
| L(s) = 1 | + 0.668·3-s + (0.321 − 0.946i)5-s + 0.669i·7-s − 0.552·9-s − 0.644i·11-s − 0.763·13-s + (0.215 − 0.633i)15-s + 1.70i·17-s − 0.667i·19-s + 0.447i·21-s + 1.57i·23-s + (−0.792 − 0.609i)25-s − 1.03·27-s + 0.715i·29-s − 0.185·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.441 - 0.897i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.441 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.060328986\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.060328986\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-17.9 + 52.9i)T \) |
| good | 3 | \( 1 - 10.4T + 243T^{2} \) |
| 7 | \( 1 - 86.7iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 258. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 465.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.02e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 1.05e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 3.99e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 3.24e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 991.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 8.82e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 5.63e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.77e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.77e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 2.59e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.81e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 3.54e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 4.20e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.13e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.25e3iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 8.42e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.90e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.51e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.39e4iT - 8.58e9T^{2} \) |
| 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
−11.17303949548038027157446438820, −9.940042538050159097263702381869, −8.900501306034852865217346816036, −8.623841518584268144180265921490, −7.52183039864208070592866918411, −5.93484364142283190825576368630, −5.31416854926881078999906341050, −3.86981461302653905169192424879, −2.63586021042387483919498315591, −1.47091482650619724902993193984,
0.23327935948299229808759598813, 2.19538172380811106092459178259, 2.94627807462231791120575563980, 4.23119117083858640639027041970, 5.55561471689188191014375907591, 6.92336345191099015666111553551, 7.43374106549567326518820430800, 8.620034663726760505675910442709, 9.732919368660450515801109015957, 10.28705339272330739499908650335
