| L(s) = 1 | + (1.02e3 + 592. i)5-s + (2.36e3 − 386. i)7-s + (−1.80e4 + 1.03e4i)11-s + 2.00e4·13-s + (4.81e4 − 2.77e4i)17-s + (1.04e5 − 1.81e5i)19-s + (1.94e5 + 1.12e5i)23-s + (5.05e5 + 8.76e5i)25-s − 6.05e5i·29-s + (8.23e5 + 1.42e6i)31-s + (2.65e6 + 1.00e6i)35-s + (1.81e5 − 3.14e5i)37-s − 5.07e6i·41-s + 1.81e6·43-s + (−2.07e6 − 1.19e6i)47-s + ⋯ |
| L(s) = 1 | + (1.64 + 0.947i)5-s + (0.986 − 0.160i)7-s + (−1.22 + 0.710i)11-s + 0.700·13-s + (0.576 − 0.332i)17-s + (0.804 − 1.39i)19-s + (0.694 + 0.400i)23-s + (1.29 + 2.24i)25-s − 0.855i·29-s + (0.891 + 1.54i)31-s + (1.77 + 0.671i)35-s + (0.0969 − 0.167i)37-s − 1.79i·41-s + 0.530·43-s + (−0.425 − 0.245i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.792 - 0.610i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.792 - 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(3.865095097\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.865095097\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.36e3 + 386. i)T \) |
| good | 5 | \( 1 + (-1.02e3 - 592. i)T + (1.95e5 + 3.38e5i)T^{2} \) |
| 11 | \( 1 + (1.80e4 - 1.03e4i)T + (1.07e8 - 1.85e8i)T^{2} \) |
| 13 | \( 1 - 2.00e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + (-4.81e4 + 2.77e4i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-1.04e5 + 1.81e5i)T + (-8.49e9 - 1.47e10i)T^{2} \) |
| 23 | \( 1 + (-1.94e5 - 1.12e5i)T + (3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 + 6.05e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + (-8.23e5 - 1.42e6i)T + (-4.26e11 + 7.38e11i)T^{2} \) |
| 37 | \( 1 + (-1.81e5 + 3.14e5i)T + (-1.75e12 - 3.04e12i)T^{2} \) |
| 41 | \( 1 + 5.07e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 1.81e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + (2.07e6 + 1.19e6i)T + (1.19e13 + 2.06e13i)T^{2} \) |
| 53 | \( 1 + (9.71e6 - 5.60e6i)T + (3.11e13 - 5.39e13i)T^{2} \) |
| 59 | \( 1 + (2.98e5 - 1.72e5i)T + (7.34e13 - 1.27e14i)T^{2} \) |
| 61 | \( 1 + (-6.29e6 + 1.09e7i)T + (-9.58e13 - 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-3.60e6 - 6.23e6i)T + (-2.03e14 + 3.51e14i)T^{2} \) |
| 71 | \( 1 - 2.09e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + (1.81e7 + 3.13e7i)T + (-4.03e14 + 6.98e14i)T^{2} \) |
| 79 | \( 1 + (9.31e5 - 1.61e6i)T + (-7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 - 7.69e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (5.74e7 + 3.31e7i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 - 1.99e7T + 7.83e15T^{2} \) |
| 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
−10.63538529508671188562320987101, −9.923290941368609063994563024622, −8.947938892061302673969927836335, −7.59072678864568087781790798861, −6.81121779576259750646896127143, −5.54350888923429901992678353740, −4.93522804019070512678142838845, −3.02620486851916586254593370664, −2.20147078594769597443379392870, −1.10709205428437557890689121426,
0.964542106533425326142930808040, 1.66431813422184860200349895186, 2.90493556258653405460683399585, 4.67813448142586648972082116577, 5.54889026228693531973522658990, 6.07201159860152526085317525313, 7.951873402687223639048231461127, 8.482766884871393883543946925399, 9.619591435135853703488153754397, 10.37336958756976054522335395743
