| L(s) = 1 | − 4·2-s + 9·3-s + 16·4-s − 25·5-s − 36·6-s − 82·7-s − 64·8-s + 81·9-s + 100·10-s + 228·11-s + 144·12-s − 232·13-s + 328·14-s − 225·15-s + 256·16-s − 546·17-s − 324·18-s + 361·19-s − 400·20-s − 738·21-s − 912·22-s + 2.62e3·23-s − 576·24-s + 625·25-s + 928·26-s + 729·27-s − 1.31e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.632·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.568·11-s + 0.288·12-s − 0.380·13-s + 0.447·14-s − 0.258·15-s + 1/4·16-s − 0.458·17-s − 0.235·18-s + 0.229·19-s − 0.223·20-s − 0.365·21-s − 0.401·22-s + 1.03·23-s − 0.204·24-s + 1/5·25-s + 0.269·26-s + 0.192·27-s − 0.316·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{2} T \) |
| 3 | \( 1 - p^{2} T \) |
| 5 | \( 1 + p^{2} T \) |
| 19 | \( 1 - p^{2} T \) |
| good | 7 | \( 1 + 82 T + p^{5} T^{2} \) |
| 11 | \( 1 - 228 T + p^{5} T^{2} \) |
| 13 | \( 1 + 232 T + p^{5} T^{2} \) |
| 17 | \( 1 + 546 T + p^{5} T^{2} \) |
| 23 | \( 1 - 114 p T + p^{5} T^{2} \) |
| 29 | \( 1 + 3222 T + p^{5} T^{2} \) |
| 31 | \( 1 - 5408 T + p^{5} T^{2} \) |
| 37 | \( 1 + 112 T + p^{5} T^{2} \) |
| 41 | \( 1 + 8946 T + p^{5} T^{2} \) |
| 43 | \( 1 - 10730 T + p^{5} T^{2} \) |
| 47 | \( 1 - 14478 T + p^{5} T^{2} \) |
| 53 | \( 1 - 1044 T + p^{5} T^{2} \) |
| 59 | \( 1 + 46284 T + p^{5} T^{2} \) |
| 61 | \( 1 + 44506 T + p^{5} T^{2} \) |
| 67 | \( 1 - 21260 T + p^{5} T^{2} \) |
| 71 | \( 1 - 61560 T + p^{5} T^{2} \) |
| 73 | \( 1 + 87154 T + p^{5} T^{2} \) |
| 79 | \( 1 + 57688 T + p^{5} T^{2} \) |
| 83 | \( 1 + 36690 T + p^{5} T^{2} \) |
| 89 | \( 1 - 92190 T + p^{5} T^{2} \) |
| 97 | \( 1 - 114692 T + p^{5} T^{2} \) |
| 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
−9.316923774636893699228366672786, −8.833335504967253097706548943167, −7.76186099259377790043902215479, −7.05125407038851896381190949123, −6.16689072634080157655175701443, −4.68857142713491090028393257648, −3.51707205777771022096634207132, −2.60944251655520709456046875400, −1.26335388048952358841823268871, 0,
1.26335388048952358841823268871, 2.60944251655520709456046875400, 3.51707205777771022096634207132, 4.68857142713491090028393257648, 6.16689072634080157655175701443, 7.05125407038851896381190949123, 7.76186099259377790043902215479, 8.833335504967253097706548943167, 9.316923774636893699228366672786
