L(s) = 1 | + (214. + 291. i)2-s − 1.63e3i·3-s + (−3.93e4 + 1.25e5i)4-s − 1.21e6i·5-s + (4.78e5 − 3.50e5i)6-s + 1.76e7·7-s + (−4.49e7 + 1.52e7i)8-s + 1.26e8·9-s + (3.53e8 − 2.59e8i)10-s − 7.26e8i·11-s + (2.04e8 + 6.45e7i)12-s + 3.16e9i·13-s + (3.78e9 + 5.15e9i)14-s − 1.98e9·15-s + (−1.40e10 − 9.84e9i)16-s + 4.56e10·17-s + ⋯ |
L(s) = 1 | + (0.591 + 0.806i)2-s − 0.144i·3-s + (−0.300 + 0.953i)4-s − 1.38i·5-s + (0.116 − 0.0852i)6-s + 1.15·7-s + (−0.946 + 0.321i)8-s + 0.979·9-s + (1.11 − 0.821i)10-s − 1.02i·11-s + (0.137 + 0.0433i)12-s + 1.07i·13-s + (0.684 + 0.933i)14-s − 0.200·15-s + (−0.819 − 0.573i)16-s + 1.58·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 - 0.321i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (0.946 - 0.321i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(9)\) |
\(\approx\) |
\(2.80285 + 0.463457i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.80285 + 0.463457i\) |
\(L(\frac{19}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-214. - 291. i)T \) |
good | 3 | \( 1 + 1.63e3iT - 1.29e8T^{2} \) |
| 5 | \( 1 + 1.21e6iT - 7.62e11T^{2} \) |
| 7 | \( 1 - 1.76e7T + 2.32e14T^{2} \) |
| 11 | \( 1 + 7.26e8iT - 5.05e17T^{2} \) |
| 13 | \( 1 - 3.16e9iT - 8.65e18T^{2} \) |
| 17 | \( 1 - 4.56e10T + 8.27e20T^{2} \) |
| 19 | \( 1 - 1.55e10iT - 5.48e21T^{2} \) |
| 23 | \( 1 - 4.09e11T + 1.41e23T^{2} \) |
| 29 | \( 1 + 3.11e12iT - 7.25e24T^{2} \) |
| 31 | \( 1 + 6.57e12T + 2.25e25T^{2} \) |
| 37 | \( 1 + 1.36e13iT - 4.56e26T^{2} \) |
| 41 | \( 1 + 2.81e12T + 2.61e27T^{2} \) |
| 43 | \( 1 - 1.36e13iT - 5.87e27T^{2} \) |
| 47 | \( 1 - 6.84e11T + 2.66e28T^{2} \) |
| 53 | \( 1 - 2.47e14iT - 2.05e29T^{2} \) |
| 59 | \( 1 - 2.03e15iT - 1.27e30T^{2} \) |
| 61 | \( 1 + 8.92e14iT - 2.24e30T^{2} \) |
| 67 | \( 1 - 5.06e15iT - 1.10e31T^{2} \) |
| 71 | \( 1 + 6.85e15T + 2.96e31T^{2} \) |
| 73 | \( 1 - 1.37e15T + 4.74e31T^{2} \) |
| 79 | \( 1 + 1.78e16T + 1.81e32T^{2} \) |
| 83 | \( 1 + 1.48e15iT - 4.21e32T^{2} \) |
| 89 | \( 1 + 3.78e16T + 1.37e33T^{2} \) |
| 97 | \( 1 - 1.28e17T + 5.95e33T^{2} \) |
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
−16.99965950796119684063262926226, −16.19046681719199722145904300627, −14.44946467525105132783946737515, −13.10569203900022895362774629896, −11.80828628978965990945765676568, −8.937628343226631797641346645253, −7.64516827751068327876161195192, −5.46108413888062433498790504866, −4.21246567400417795451611041452, −1.22236482801359392616534505069,
1.55303549721333137406206445400, 3.26197578022000518802863522119, 5.06729710721355264964753170359, 7.27188846027929701636748326058, 10.04526349373719668620134977526, 10.99749764604405298783714336737, 12.63776667942222238467590938079, 14.51045013819637865408031993025, 15.11654949181115693012658063703, 17.93864695660536674749189565311