| L(s) = 1 | + (−7.88 − 8.11i)2-s + (−32.5 − 33.6i)3-s + (−3.74 + 127. i)4-s + (−80.0 − 138. i)5-s + (−16.4 + 528. i)6-s + (−1.43e3 − 829. i)7-s + (1.06e3 − 978. i)8-s + (−71.9 + 2.18e3i)9-s + (−494. + 1.74e3i)10-s + (−5.09e3 − 2.94e3i)11-s + (4.42e3 − 4.03e3i)12-s + (−3.70e3 + 2.13e3i)13-s + (4.59e3 + 1.82e4i)14-s + (−2.05e3 + 7.20e3i)15-s + (−1.63e4 − 957. i)16-s + 9.86e3i·17-s + ⋯ |
| L(s) = 1 | + (−0.696 − 0.717i)2-s + (−0.695 − 0.718i)3-s + (−0.0292 + 0.999i)4-s + (−0.286 − 0.496i)5-s + (−0.0310 + 0.999i)6-s + (−1.58 − 0.914i)7-s + (0.737 − 0.675i)8-s + (−0.0329 + 0.999i)9-s + (−0.156 + 0.551i)10-s + (−1.15 − 0.666i)11-s + (0.738 − 0.674i)12-s + (−0.467 + 0.270i)13-s + (0.447 + 1.77i)14-s + (−0.157 + 0.550i)15-s + (−0.998 − 0.0584i)16-s + 0.487i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0104i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0104i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0147086 + 7.70273\times10^{-5}i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0147086 + 7.70273\times10^{-5}i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (7.88 + 8.11i)T \) |
| 3 | \( 1 + (32.5 + 33.6i)T \) |
| good | 5 | \( 1 + (80.0 + 138. i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 7 | \( 1 + (1.43e3 + 829. i)T + (4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (5.09e3 + 2.94e3i)T + (9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + (3.70e3 - 2.13e3i)T + (3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 - 9.86e3iT - 4.10e8T^{2} \) |
| 19 | \( 1 + 4.10e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + (1.15e4 + 2.00e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-2.39e4 + 4.14e4i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 + (-1.49e5 + 8.62e4i)T + (1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 + 5.60e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 + (2.80e5 - 1.61e5i)T + (9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (-1.62e5 + 2.81e5i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 + (1.26e5 - 2.18e5i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 - 1.87e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + (6.19e5 - 3.57e5i)T + (1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (1.44e6 + 8.35e5i)T + (1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-3.83e5 - 6.64e5i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + 3.84e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.36e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + (-8.32e5 - 4.80e5i)T + (9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (1.82e6 + 1.05e6i)T + (1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 - 7.79e6iT - 4.42e13T^{2} \) |
| 97 | \( 1 + (3.08e6 - 5.33e6i)T + (-4.03e13 - 6.99e13i)T^{2} \) |
| 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
−12.89877492062463845654923924356, −12.22891894005607681068210155984, −10.76392327464819343607250496401, −10.16618529510921941471208555384, −8.566122426925945523456668578894, −7.43140503497603536544039379907, −6.26787985097181505633701207055, −4.22315266512426749807886344603, −2.53008810946892171113341853894, −0.57158320646139783780577385916,
0.01343456414117775890723047350, 2.84658997855557455843103078085, 4.93324123788702509651099208327, 6.11974707839554182182218496190, 7.06690559003707225688952523796, 8.756542561134382511466870935865, 9.928808978060354043595443437376, 10.43132221830992366789452109234, 11.93644081381318597906930718133, 13.12079911425744900363369716275
