| L(s) = 1 | − 27i·3-s + 218. i·5-s + 904.·7-s − 729·9-s + 1.95e3i·11-s + 1.49e3i·13-s + 5.90e3·15-s + 1.46e4·17-s + 3.25e4i·19-s − 2.44e4i·21-s − 2.42e4·23-s + 3.02e4·25-s + 1.96e4i·27-s − 4.77e4i·29-s − 3.84e4·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.783i·5-s + 0.996·7-s − 0.333·9-s + 0.443i·11-s + 0.188i·13-s + 0.452·15-s + 0.723·17-s + 1.08i·19-s − 0.575i·21-s − 0.415·23-s + 0.386·25-s + 0.192i·27-s − 0.363i·29-s − 0.231·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.993246096\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.993246096\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 27iT \) |
| good | 5 | \( 1 - 218. iT - 7.81e4T^{2} \) |
| 7 | \( 1 - 904.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 1.95e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 - 1.49e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 - 1.46e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.25e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 + 2.42e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 4.77e4iT - 1.72e10T^{2} \) |
| 31 | \( 1 + 3.84e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.30e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 - 5.03e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 7.41e4iT - 2.71e11T^{2} \) |
| 47 | \( 1 + 8.87e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.01e6iT - 1.17e12T^{2} \) |
| 59 | \( 1 + 9.56e5iT - 2.48e12T^{2} \) |
| 61 | \( 1 + 1.51e6iT - 3.14e12T^{2} \) |
| 67 | \( 1 + 2.12e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 - 1.47e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.76e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.66e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 6.45e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 - 3.18e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.36e7T + 8.07e13T^{2} \) |
| 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
−10.48509565959920865327192237974, −9.552453227974702043935884409036, −8.206452752423153191535698928969, −7.68514690090008447562596922859, −6.69240569700915201276176878433, −5.73744465549573845976757081580, −4.57458801948195148495968217661, −3.29661229603868959861226593205, −2.10265879030147209007247627306, −1.20765530163093543598419337961,
0.42331050621680193658613247190, 1.49511638232391986078750934596, 2.91630951101197439827724190446, 4.21224453239208088171861337470, 5.01693988151903907574006277754, 5.78353877935740085631702236755, 7.27174455841009409135619576937, 8.313578149039313270578985868604, 8.895907412113442733883801370447, 9.888376916427682268979869874580
