| L(s) = 1 | − 2-s − 3·3-s − 7·4-s + 3·6-s − 36·7-s + 15·8-s + 9·9-s + 11·11-s + 21·12-s − 2·13-s + 36·14-s + 41·16-s − 66·17-s − 9·18-s + 140·19-s + 108·21-s − 11·22-s + 68·23-s − 45·24-s + 2·26-s − 27·27-s + 252·28-s + 150·29-s − 128·31-s − 161·32-s − 33·33-s + 66·34-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 0.577·3-s − 7/8·4-s + 0.204·6-s − 1.94·7-s + 0.662·8-s + 1/3·9-s + 0.301·11-s + 0.505·12-s − 0.0426·13-s + 0.687·14-s + 0.640·16-s − 0.941·17-s − 0.117·18-s + 1.69·19-s + 1.12·21-s − 0.106·22-s + 0.616·23-s − 0.382·24-s + 0.0150·26-s − 0.192·27-s + 1.70·28-s + 0.960·29-s − 0.741·31-s − 0.889·32-s − 0.174·33-s + 0.332·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - p T \) |
| good | 2 | \( 1 + T + p^{3} T^{2} \) |
| 7 | \( 1 + 36 T + p^{3} T^{2} \) |
| 13 | \( 1 + 2 T + p^{3} T^{2} \) |
| 17 | \( 1 + 66 T + p^{3} T^{2} \) |
| 19 | \( 1 - 140 T + p^{3} T^{2} \) |
| 23 | \( 1 - 68 T + p^{3} T^{2} \) |
| 29 | \( 1 - 150 T + p^{3} T^{2} \) |
| 31 | \( 1 + 128 T + p^{3} T^{2} \) |
| 37 | \( 1 - 314 T + p^{3} T^{2} \) |
| 41 | \( 1 + 118 T + p^{3} T^{2} \) |
| 43 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 47 | \( 1 - 324 T + p^{3} T^{2} \) |
| 53 | \( 1 + 82 T + p^{3} T^{2} \) |
| 59 | \( 1 + 740 T + p^{3} T^{2} \) |
| 61 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 67 | \( 1 - 124 T + p^{3} T^{2} \) |
| 71 | \( 1 + 988 T + p^{3} T^{2} \) |
| 73 | \( 1 + 2 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1100 T + p^{3} T^{2} \) |
| 83 | \( 1 - 868 T + p^{3} T^{2} \) |
| 89 | \( 1 + 470 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1186 T + p^{3} T^{2} \) |
| 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
−9.420939910550260855737171750095, −8.951775830981159239088584618965, −7.60883185051347564951658221118, −6.77188944019531345156763487308, −5.96764016112076086723829471674, −4.96422622703958747464573790397, −3.89585764858646310557682423135, −2.95260744985414853091495640129, −0.989585998782977495469420025380, 0,
0.989585998782977495469420025380, 2.95260744985414853091495640129, 3.89585764858646310557682423135, 4.96422622703958747464573790397, 5.96764016112076086723829471674, 6.77188944019531345156763487308, 7.60883185051347564951658221118, 8.951775830981159239088584618965, 9.420939910550260855737171750095
