L(s) = 1 | + (40.5 − 23.3i)3-s + (−133. − 77.3i)5-s + (−2.23e3 + 877. i)7-s + (1.09e3 − 1.89e3i)9-s + (3.95e3 + 6.84e3i)11-s + 4.17e3i·13-s − 7.23e3·15-s + (1.10e5 − 6.38e4i)17-s + (1.39e5 + 8.03e4i)19-s + (−6.99e4 + 8.77e4i)21-s + (−1.64e4 + 2.84e4i)23-s + (−1.83e5 − 3.17e5i)25-s − 1.02e5i·27-s + 6.59e5·29-s + (−5.16e3 + 2.98e3i)31-s + ⋯ |
L(s) = 1 | + (0.5 − 0.288i)3-s + (−0.214 − 0.123i)5-s + (−0.930 + 0.365i)7-s + (0.166 − 0.288i)9-s + (0.269 + 0.467i)11-s + 0.146i·13-s − 0.142·15-s + (1.32 − 0.764i)17-s + (1.06 + 0.616i)19-s + (−0.359 + 0.451i)21-s + (−0.0586 + 0.101i)23-s + (−0.469 − 0.813i)25-s − 0.192i·27-s + 0.932·29-s + (−0.00559 + 0.00322i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.272i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.962 + 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(2.15898 - 0.300046i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.15898 - 0.300046i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-40.5 + 23.3i)T \) |
| 7 | \( 1 + (2.23e3 - 877. i)T \) |
good | 5 | \( 1 + (133. + 77.3i)T + (1.95e5 + 3.38e5i)T^{2} \) |
| 11 | \( 1 + (-3.95e3 - 6.84e3i)T + (-1.07e8 + 1.85e8i)T^{2} \) |
| 13 | \( 1 - 4.17e3iT - 8.15e8T^{2} \) |
| 17 | \( 1 + (-1.10e5 + 6.38e4i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-1.39e5 - 8.03e4i)T + (8.49e9 + 1.47e10i)T^{2} \) |
| 23 | \( 1 + (1.64e4 - 2.84e4i)T + (-3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 - 6.59e5T + 5.00e11T^{2} \) |
| 31 | \( 1 + (5.16e3 - 2.98e3i)T + (4.26e11 - 7.38e11i)T^{2} \) |
| 37 | \( 1 + (-3.29e5 + 5.70e5i)T + (-1.75e12 - 3.04e12i)T^{2} \) |
| 41 | \( 1 + 1.10e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 4.57e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + (1.35e6 + 7.84e5i)T + (1.19e13 + 2.06e13i)T^{2} \) |
| 53 | \( 1 + (-4.31e6 - 7.46e6i)T + (-3.11e13 + 5.39e13i)T^{2} \) |
| 59 | \( 1 + (-1.85e7 + 1.07e7i)T + (7.34e13 - 1.27e14i)T^{2} \) |
| 61 | \( 1 + (-2.16e7 - 1.25e7i)T + (9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (9.65e6 + 1.67e7i)T + (-2.03e14 + 3.51e14i)T^{2} \) |
| 71 | \( 1 - 9.50e6T + 6.45e14T^{2} \) |
| 73 | \( 1 + (-6.91e6 + 3.99e6i)T + (4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (1.32e6 - 2.29e6i)T + (-7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 - 5.37e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (3.33e7 + 1.92e7i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 - 2.56e7iT - 7.83e15T^{2} \) |
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
−12.42925612881154766287663640769, −11.90688520018817657055566099650, −10.06547771610212932681563025021, −9.339391253085444849136714792030, −8.004788411786501465919141789902, −6.93218427305231370759596620206, −5.57205010079561531293601325292, −3.81854939964350195386387680580, −2.60164516289879544294932819825, −0.878128059882366444452079891846,
0.932732940275572432934512714706, 2.96483844625431710978486573100, 3.86780347647707147798574219525, 5.61095334493161863459983383233, 7.01923051779239429535378208720, 8.177876101233350547635231081585, 9.469899127529163033284928887510, 10.27850410993525063303315832119, 11.58066779527270606031131202043, 12.81320082345020143943884132394