| L(s) = 1 | + 27.1i·3-s + 170.·5-s + 374. i·7-s − 10.3·9-s − 1.35e3i·11-s + 2.95e3·13-s + 4.64e3i·15-s − 6.65e3·17-s + 9.92e3i·19-s − 1.01e4·21-s + 1.55e4i·23-s + 1.35e4·25-s + 1.95e4i·27-s + 2.08e3·29-s − 2.84e4i·31-s + ⋯ |
| L(s) = 1 | + 1.00i·3-s + 1.36·5-s + 1.09i·7-s − 0.0142·9-s − 1.02i·11-s + 1.34·13-s + 1.37i·15-s − 1.35·17-s + 1.44i·19-s − 1.09·21-s + 1.28i·23-s + 0.866·25-s + 0.992i·27-s + 0.0854·29-s − 0.954i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.64941 + 1.64941i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.64941 + 1.64941i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 27.1iT - 729T^{2} \) |
| 5 | \( 1 - 170.T + 1.56e4T^{2} \) |
| 7 | \( 1 - 374. iT - 1.17e5T^{2} \) |
| 11 | \( 1 + 1.35e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 2.95e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + 6.65e3T + 2.41e7T^{2} \) |
| 19 | \( 1 - 9.92e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 - 1.55e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 2.08e3T + 5.94e8T^{2} \) |
| 31 | \( 1 + 2.84e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 3.33e3T + 2.56e9T^{2} \) |
| 41 | \( 1 + 6.31e4T + 4.75e9T^{2} \) |
| 43 | \( 1 + 8.85e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 + 1.08e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 4.80e4T + 2.21e10T^{2} \) |
| 59 | \( 1 + 1.47e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 1.24e5T + 5.15e10T^{2} \) |
| 67 | \( 1 - 8.75e4iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 2.25e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 4.75e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 2.76e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 4.85e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 1.06e6T + 4.96e11T^{2} \) |
| 97 | \( 1 - 1.46e6T + 8.32e11T^{2} \) |
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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
−13.89751122220987161310605576085, −13.13058053258834909303331905255, −11.47498402655084425583312547234, −10.40907643771719951245995890003, −9.366648915092406717057919551352, −8.582705754702405941586591291812, −6.17101055462284657098068938955, −5.45706919275599210227166595919, −3.63594598732200524043013768912, −1.85153426215052796387064821991,
1.03050019483365234879901476048, 2.21595822029121633376734108550, 4.55258915689987191278189401236, 6.45727548083119576141006315374, 7.00148098303262076359805487814, 8.692841492061888578964473173722, 10.02848198953105734220346820465, 11.03767072167604297201836866334, 12.77945942885226770497241869370, 13.37348299631728095028804967974
