| L(s) = 1 | − 4·3-s − 8·4-s + 22·7-s − 11·9-s − 12·11-s + 32·12-s − 8·13-s + 64·16-s + 66·17-s + 19·19-s − 88·21-s + 30·23-s + 152·27-s − 176·28-s − 6·29-s − 64·31-s + 48·33-s + 88·36-s + 16·37-s + 32·39-s + 54·41-s − 182·43-s + 96·44-s − 594·47-s − 256·48-s + 141·49-s − 264·51-s + ⋯ |
| L(s) = 1 | − 0.769·3-s − 4-s + 1.18·7-s − 0.407·9-s − 0.328·11-s + 0.769·12-s − 0.170·13-s + 16-s + 0.941·17-s + 0.229·19-s − 0.914·21-s + 0.271·23-s + 1.08·27-s − 1.18·28-s − 0.0384·29-s − 0.370·31-s + 0.253·33-s + 0.407·36-s + 0.0710·37-s + 0.131·39-s + 0.205·41-s − 0.645·43-s + 0.328·44-s − 1.84·47-s − 0.769·48-s + 0.411·49-s − 0.724·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 19 | \( 1 - p T \) |
| good | 2 | \( 1 + p^{3} T^{2} \) |
| 3 | \( 1 + 4 T + p^{3} T^{2} \) |
| 7 | \( 1 - 22 T + p^{3} T^{2} \) |
| 11 | \( 1 + 12 T + p^{3} T^{2} \) |
| 13 | \( 1 + 8 T + p^{3} T^{2} \) |
| 17 | \( 1 - 66 T + p^{3} T^{2} \) |
| 23 | \( 1 - 30 T + p^{3} T^{2} \) |
| 29 | \( 1 + 6 T + p^{3} T^{2} \) |
| 31 | \( 1 + 64 T + p^{3} T^{2} \) |
| 37 | \( 1 - 16 T + p^{3} T^{2} \) |
| 41 | \( 1 - 54 T + p^{3} T^{2} \) |
| 43 | \( 1 + 182 T + p^{3} T^{2} \) |
| 47 | \( 1 + 594 T + p^{3} T^{2} \) |
| 53 | \( 1 + 396 T + p^{3} T^{2} \) |
| 59 | \( 1 + 564 T + p^{3} T^{2} \) |
| 61 | \( 1 + 706 T + p^{3} T^{2} \) |
| 67 | \( 1 - 628 T + p^{3} T^{2} \) |
| 71 | \( 1 + 984 T + p^{3} T^{2} \) |
| 73 | \( 1 + 14 T + p^{3} T^{2} \) |
| 79 | \( 1 + 328 T + p^{3} T^{2} \) |
| 83 | \( 1 - 294 T + p^{3} T^{2} \) |
| 89 | \( 1 - 918 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1564 T + p^{3} T^{2} \) |
| 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
−10.24464973115880023443016256060, −9.248663479251522600889222985506, −8.269051748585683001285484734243, −7.64085579853409930559867825959, −6.12013719895665220381052905197, −5.17191996128239904701002621092, −4.71731316967654858480096756629, −3.24234439464804457591569742539, −1.36576516182228983124086926727, 0,
1.36576516182228983124086926727, 3.24234439464804457591569742539, 4.71731316967654858480096756629, 5.17191996128239904701002621092, 6.12013719895665220381052905197, 7.64085579853409930559867825959, 8.269051748585683001285484734243, 9.248663479251522600889222985506, 10.24464973115880023443016256060
