L(s) = 1 | + (15.4 + 2.11i)3-s + 68.1i·5-s + 49i·7-s + (234. + 65.2i)9-s − 164.·11-s − 181.·13-s + (−144. + 1.05e3i)15-s − 68.1i·17-s + 1.79e3i·19-s + (−103. + 756. i)21-s + 435.·23-s − 1.52e3·25-s + (3.47e3 + 1.50e3i)27-s + 1.91e3i·29-s + 3.13e3i·31-s + ⋯ |
L(s) = 1 | + (0.990 + 0.135i)3-s + 1.21i·5-s + 0.377i·7-s + (0.963 + 0.268i)9-s − 0.408·11-s − 0.297·13-s + (−0.165 + 1.20i)15-s − 0.0571i·17-s + 1.13i·19-s + (−0.0512 + 0.374i)21-s + 0.171·23-s − 0.486·25-s + (0.917 + 0.396i)27-s + 0.422i·29-s + 0.585i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.790 - 0.612i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.259766750\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.259766750\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-15.4 - 2.11i)T \) |
| 7 | \( 1 - 49iT \) |
good | 5 | \( 1 - 68.1iT - 3.12e3T^{2} \) |
| 11 | \( 1 + 164.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 181.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 68.1iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 1.79e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 435.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.91e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 3.13e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + 8.70e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 9.36e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 2.07e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 2.43e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.46e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 2.02e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.29e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.11e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 2.13e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.37e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.30e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + 9.91e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.92e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 1.76e5T + 8.58e9T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.76761134171369634378779783656, −10.22978074827858003071754787866, −9.246368454316350501580990511539, −8.220195647975320876660321148364, −7.37492023951667679004265089864, −6.48466288388219259258911457041, −5.10353181948889614654243103442, −3.64153654580878239798799617832, −2.86374607508857445557766472064, −1.79472594366885014580729577602,
0.47919186714146924426825447328, 1.68632088557104768617106408318, 2.99773054913324654483939307956, 4.32081764255604531465267643121, 5.10389767097637456418392523107, 6.68377873140733897158510373443, 7.73575929898688290946429752476, 8.492800534015804064454269982161, 9.297237119862652569952715843059, 10.06457576911440572886052829132