L(s) = 1 | + (−5.65 + 9.79i)2-s + (40.5 − 23.3i)3-s + (−63.9 − 110. i)4-s + (−271. − 156. i)5-s + 529. i·6-s + (746. + 2.28e3i)7-s + 1.44e3·8-s + (1.09e3 − 1.89e3i)9-s + (3.06e3 − 1.77e3i)10-s + (−5.33e3 − 9.24e3i)11-s + (−5.18e3 − 2.99e3i)12-s − 2.08e4i·13-s + (−2.65e4 − 5.59e3i)14-s − 1.46e4·15-s + (−8.19e3 + 1.41e4i)16-s + (5.47e4 − 3.16e4i)17-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.5 − 0.288i)3-s + (−0.249 − 0.433i)4-s + (−0.433 − 0.250i)5-s + 0.408i·6-s + (0.310 + 0.950i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (0.306 − 0.177i)10-s + (−0.364 − 0.631i)11-s + (−0.249 − 0.144i)12-s − 0.730i·13-s + (−0.691 − 0.145i)14-s − 0.289·15-s + (−0.125 + 0.216i)16-s + (0.655 − 0.378i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 + 0.756i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.654 + 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(1.29130 - 0.590531i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.29130 - 0.590531i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (5.65 - 9.79i)T \) |
| 3 | \( 1 + (-40.5 + 23.3i)T \) |
| 7 | \( 1 + (-746. - 2.28e3i)T \) |
good | 5 | \( 1 + (271. + 156. i)T + (1.95e5 + 3.38e5i)T^{2} \) |
| 11 | \( 1 + (5.33e3 + 9.24e3i)T + (-1.07e8 + 1.85e8i)T^{2} \) |
| 13 | \( 1 + 2.08e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + (-5.47e4 + 3.16e4i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (7.96e4 + 4.59e4i)T + (8.49e9 + 1.47e10i)T^{2} \) |
| 23 | \( 1 + (-2.45e5 + 4.24e5i)T + (-3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 - 1.10e6T + 5.00e11T^{2} \) |
| 31 | \( 1 + (-9.52e5 + 5.50e5i)T + (4.26e11 - 7.38e11i)T^{2} \) |
| 37 | \( 1 + (5.38e5 - 9.33e5i)T + (-1.75e12 - 3.04e12i)T^{2} \) |
| 41 | \( 1 + 7.98e5iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 2.93e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + (3.93e6 + 2.26e6i)T + (1.19e13 + 2.06e13i)T^{2} \) |
| 53 | \( 1 + (1.45e6 + 2.51e6i)T + (-3.11e13 + 5.39e13i)T^{2} \) |
| 59 | \( 1 + (-1.25e7 + 7.25e6i)T + (7.34e13 - 1.27e14i)T^{2} \) |
| 61 | \( 1 + (8.82e6 + 5.09e6i)T + (9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-1.40e7 - 2.43e7i)T + (-2.03e14 + 3.51e14i)T^{2} \) |
| 71 | \( 1 + 2.65e7T + 6.45e14T^{2} \) |
| 73 | \( 1 + (4.39e6 - 2.53e6i)T + (4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (6.92e6 - 1.20e7i)T + (-7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 + 4.69e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (-1.96e7 - 1.13e7i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 - 1.32e8iT - 7.83e15T^{2} \) |
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
−14.36713026626626390761904141955, −13.03033246014264068656769837297, −11.83799849985254956093462648201, −10.22482859225315041860330689761, −8.595154610963579954992042027078, −8.138167116164210301602865039067, −6.43556618497666260978977456496, −4.91542765774100653373474672933, −2.73144798370055543249755542046, −0.62075137141345265999490784876,
1.48329423374438862053317534778, 3.33145433949097105435946120270, 4.58323329201547885768328480838, 7.15661182098531432252250411036, 8.225633214875551485147981628145, 9.722805066504597092459385734451, 10.66918814735313080430236019738, 11.85576947347680052766613616393, 13.27097362743432346545847409564, 14.33923689759300374938252526751