| L(s) = 1 | + 3·3-s − 7·5-s + 9·9-s + 7·11-s + 52·13-s − 21·15-s − 72·17-s − 20·19-s − 48·23-s − 76·25-s + 27·27-s − 243·29-s − 95·31-s + 21·33-s + 352·37-s + 156·39-s + 296·41-s + 158·43-s − 63·45-s + 142·47-s − 216·51-s − 375·53-s − 49·55-s − 60·57-s − 279·59-s − 246·61-s − 364·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.626·5-s + 1/3·9-s + 0.191·11-s + 1.10·13-s − 0.361·15-s − 1.02·17-s − 0.241·19-s − 0.435·23-s − 0.607·25-s + 0.192·27-s − 1.55·29-s − 0.550·31-s + 0.110·33-s + 1.56·37-s + 0.640·39-s + 1.12·41-s + 0.560·43-s − 0.208·45-s + 0.440·47-s − 0.593·51-s − 0.971·53-s − 0.120·55-s − 0.139·57-s − 0.615·59-s − 0.516·61-s − 0.694·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 7 T + p^{3} T^{2} \) |
| 11 | \( 1 - 7 T + p^{3} T^{2} \) |
| 13 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 17 | \( 1 + 72 T + p^{3} T^{2} \) |
| 19 | \( 1 + 20 T + p^{3} T^{2} \) |
| 23 | \( 1 + 48 T + p^{3} T^{2} \) |
| 29 | \( 1 + 243 T + p^{3} T^{2} \) |
| 31 | \( 1 + 95 T + p^{3} T^{2} \) |
| 37 | \( 1 - 352 T + p^{3} T^{2} \) |
| 41 | \( 1 - 296 T + p^{3} T^{2} \) |
| 43 | \( 1 - 158 T + p^{3} T^{2} \) |
| 47 | \( 1 - 142 T + p^{3} T^{2} \) |
| 53 | \( 1 + 375 T + p^{3} T^{2} \) |
| 59 | \( 1 + 279 T + p^{3} T^{2} \) |
| 61 | \( 1 + 246 T + p^{3} T^{2} \) |
| 67 | \( 1 + 730 T + p^{3} T^{2} \) |
| 71 | \( 1 - 338 T + p^{3} T^{2} \) |
| 73 | \( 1 - 542 T + p^{3} T^{2} \) |
| 79 | \( 1 + 305 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1123 T + p^{3} T^{2} \) |
| 89 | \( 1 - 426 T + p^{3} T^{2} \) |
| 97 | \( 1 - 369 T + p^{3} T^{2} \) |
| 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
−9.064224336450599816117680569799, −8.081156442447741886020100544396, −7.57778987601762511570753995390, −6.50820019080113052756356762563, −5.70075677241794638342053421909, −4.24109403655133679862600668425, −3.87454242501765077009975674973, −2.63738095008052658831454637358, −1.48839103619666642990200602945, 0,
1.48839103619666642990200602945, 2.63738095008052658831454637358, 3.87454242501765077009975674973, 4.24109403655133679862600668425, 5.70075677241794638342053421909, 6.50820019080113052756356762563, 7.57778987601762511570753995390, 8.081156442447741886020100544396, 9.064224336450599816117680569799
