| L(s) = 1 | + (−7.10 − 14.3i)2-s + (−91.4 − 37.8i)3-s + (−154. + 203. i)4-s + (833. − 345. i)5-s + (107. + 1.58e3i)6-s + (1.94e3 + 1.94e3i)7-s + (4.02e3 + 772. i)8-s + (2.28e3 + 2.28e3i)9-s + (−1.08e4 − 9.49e3i)10-s + (4.42e3 − 1.83e3i)11-s + (2.18e4 − 1.27e4i)12-s + (1.42e4 + 5.89e3i)13-s + (1.40e4 − 4.16e4i)14-s − 8.93e4·15-s + (−1.75e4 − 6.31e4i)16-s − 9.72e4i·17-s + ⋯ |
| L(s) = 1 | + (−0.444 − 0.895i)2-s + (−1.12 − 0.467i)3-s + (−0.605 + 0.796i)4-s + (1.33 − 0.552i)5-s + (0.0826 + 1.21i)6-s + (0.809 + 0.809i)7-s + (0.982 + 0.188i)8-s + (0.349 + 0.349i)9-s + (−1.08 − 0.949i)10-s + (0.301 − 0.125i)11-s + (1.05 − 0.615i)12-s + (0.498 + 0.206i)13-s + (0.365 − 1.08i)14-s − 1.76·15-s + (−0.267 − 0.963i)16-s − 1.16i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.248 + 0.968i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.248 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.766548 - 0.987736i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.766548 - 0.987736i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (7.10 + 14.3i)T \) |
| good | 3 | \( 1 + (91.4 + 37.8i)T + (4.63e3 + 4.63e3i)T^{2} \) |
| 5 | \( 1 + (-833. + 345. i)T + (2.76e5 - 2.76e5i)T^{2} \) |
| 7 | \( 1 + (-1.94e3 - 1.94e3i)T + 5.76e6iT^{2} \) |
| 11 | \( 1 + (-4.42e3 + 1.83e3i)T + (1.51e8 - 1.51e8i)T^{2} \) |
| 13 | \( 1 + (-1.42e4 - 5.89e3i)T + (5.76e8 + 5.76e8i)T^{2} \) |
| 17 | \( 1 + 9.72e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + (8.79e4 - 2.12e5i)T + (-1.20e10 - 1.20e10i)T^{2} \) |
| 23 | \( 1 + (-3.28e5 + 3.28e5i)T - 7.83e10iT^{2} \) |
| 29 | \( 1 + (-4.05e5 + 9.78e5i)T + (-3.53e11 - 3.53e11i)T^{2} \) |
| 31 | \( 1 + 5.75e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 + (-2.20e5 + 9.13e4i)T + (2.48e12 - 2.48e12i)T^{2} \) |
| 41 | \( 1 + (-1.64e6 - 1.64e6i)T + 7.98e12iT^{2} \) |
| 43 | \( 1 + (1.63e6 - 6.75e5i)T + (8.26e12 - 8.26e12i)T^{2} \) |
| 47 | \( 1 - 3.69e6T + 2.38e13T^{2} \) |
| 53 | \( 1 + (3.96e6 + 9.58e6i)T + (-4.40e13 + 4.40e13i)T^{2} \) |
| 59 | \( 1 + (6.46e6 + 1.56e7i)T + (-1.03e14 + 1.03e14i)T^{2} \) |
| 61 | \( 1 + (-1.34e6 + 3.24e6i)T + (-1.35e14 - 1.35e14i)T^{2} \) |
| 67 | \( 1 + (-7.75e6 - 3.21e6i)T + (2.87e14 + 2.87e14i)T^{2} \) |
| 71 | \( 1 + (9.90e6 + 9.90e6i)T + 6.45e14iT^{2} \) |
| 73 | \( 1 + (-8.39e6 - 8.39e6i)T + 8.06e14iT^{2} \) |
| 79 | \( 1 - 3.71e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + (2.60e7 - 6.27e7i)T + (-1.59e15 - 1.59e15i)T^{2} \) |
| 89 | \( 1 + (2.14e7 - 2.14e7i)T - 3.93e15iT^{2} \) |
| 97 | \( 1 - 7.73e7T + 7.83e15T^{2} \) |
| 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
−14.25990417522167204072129075337, −12.97332340499937791366484304672, −12.02625211061005627202221690044, −11.07969854434690798510306679681, −9.644246512508253510960877578358, −8.435875100463474379417403869848, −6.18460673569397767837196094017, −4.95080393293702054296302744327, −2.10381914602104405575040182010, −0.884899706869744202153292522023,
1.24660355634998613271625567975, 4.75318430636017629999437448997, 5.89870994441353404437594014618, 7.00807178325340232794541031869, 8.978928793283270655154704897587, 10.55239919957409537300657507675, 10.85379083303688668176038071242, 13.31141757389087445325599934678, 14.32459986566689142894288164509, 15.50498599884817886369243688247
