| L(s) = 1 | + 2·2-s − 9·3-s − 28·4-s − 98·5-s − 18·6-s + 240·7-s − 120·8-s + 81·9-s − 196·10-s + 336·11-s + 252·12-s − 342·13-s + 480·14-s + 882·15-s + 656·16-s − 6·17-s + 162·18-s + 2.74e3·20-s − 2.16e3·21-s + 672·22-s + 2.83e3·23-s + 1.08e3·24-s + 6.47e3·25-s − 684·26-s − 729·27-s − 6.72e3·28-s + 5.90e3·29-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 0.577·3-s − 7/8·4-s − 1.75·5-s − 0.204·6-s + 1.85·7-s − 0.662·8-s + 1/3·9-s − 0.619·10-s + 0.837·11-s + 0.505·12-s − 0.561·13-s + 0.654·14-s + 1.01·15-s + 0.640·16-s − 0.00503·17-s + 0.117·18-s + 1.53·20-s − 1.06·21-s + 0.296·22-s + 1.11·23-s + 0.382·24-s + 2.07·25-s − 0.198·26-s − 0.192·27-s − 1.61·28-s + 1.30·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.238458613\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.238458613\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p^{2} T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - p T + p^{5} T^{2} \) |
| 5 | \( 1 + 98 T + p^{5} T^{2} \) |
| 7 | \( 1 - 240 T + p^{5} T^{2} \) |
| 11 | \( 1 - 336 T + p^{5} T^{2} \) |
| 13 | \( 1 + 342 T + p^{5} T^{2} \) |
| 17 | \( 1 + 6 T + p^{5} T^{2} \) |
| 23 | \( 1 - 2836 T + p^{5} T^{2} \) |
| 29 | \( 1 - 5902 T + p^{5} T^{2} \) |
| 31 | \( 1 + 2744 T + p^{5} T^{2} \) |
| 37 | \( 1 + 13670 T + p^{5} T^{2} \) |
| 41 | \( 1 + 10990 T + p^{5} T^{2} \) |
| 43 | \( 1 + 4996 T + p^{5} T^{2} \) |
| 47 | \( 1 + 17124 T + p^{5} T^{2} \) |
| 53 | \( 1 - 4470 T + p^{5} T^{2} \) |
| 59 | \( 1 + 26292 T + p^{5} T^{2} \) |
| 61 | \( 1 - 29134 T + p^{5} T^{2} \) |
| 67 | \( 1 - 42052 T + p^{5} T^{2} \) |
| 71 | \( 1 - 26112 T + p^{5} T^{2} \) |
| 73 | \( 1 + 49046 T + p^{5} T^{2} \) |
| 79 | \( 1 + 79056 T + p^{5} T^{2} \) |
| 83 | \( 1 - 9472 T + p^{5} T^{2} \) |
| 89 | \( 1 + 82894 T + p^{5} T^{2} \) |
| 97 | \( 1 + 39850 T + p^{5} T^{2} \) |
| 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
−8.703626824633694738070825852547, −8.464668174511156418493027699271, −7.52475269541324408661323369301, −6.79171862315968840480457350949, −5.22685908809963636955742983833, −4.80322911540794895505122688570, −4.17692089018126039515936737384, −3.28575344448757788678193431147, −1.44899326703668920145017117702, −0.51672299124882045478843576576,
0.51672299124882045478843576576, 1.44899326703668920145017117702, 3.28575344448757788678193431147, 4.17692089018126039515936737384, 4.80322911540794895505122688570, 5.22685908809963636955742983833, 6.79171862315968840480457350949, 7.52475269541324408661323369301, 8.464668174511156418493027699271, 8.703626824633694738070825852547
