| L(s) = 1 | − 5·3-s + 10·7-s − 2·9-s + 15·11-s − 8·13-s + 21·17-s − 105·19-s − 50·21-s + 10·23-s + 145·27-s − 20·29-s + 230·31-s − 75·33-s + 54·37-s + 40·39-s − 195·41-s + 300·43-s + 480·47-s − 243·49-s − 105·51-s − 322·53-s + 525·57-s − 560·59-s − 730·61-s − 20·63-s − 255·67-s − 50·69-s + ⋯ |
| L(s) = 1 | − 0.962·3-s + 0.539·7-s − 0.0740·9-s + 0.411·11-s − 0.170·13-s + 0.299·17-s − 1.26·19-s − 0.519·21-s + 0.0906·23-s + 1.03·27-s − 0.128·29-s + 1.33·31-s − 0.395·33-s + 0.239·37-s + 0.164·39-s − 0.742·41-s + 1.06·43-s + 1.48·47-s − 0.708·49-s − 0.288·51-s − 0.834·53-s + 1.21·57-s − 1.23·59-s − 1.53·61-s − 0.0399·63-s − 0.464·67-s − 0.0872·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 5 T + p^{3} T^{2} \) |
| 7 | \( 1 - 10 T + p^{3} T^{2} \) |
| 11 | \( 1 - 15 T + p^{3} T^{2} \) |
| 13 | \( 1 + 8 T + p^{3} T^{2} \) |
| 17 | \( 1 - 21 T + p^{3} T^{2} \) |
| 19 | \( 1 + 105 T + p^{3} T^{2} \) |
| 23 | \( 1 - 10 T + p^{3} T^{2} \) |
| 29 | \( 1 + 20 T + p^{3} T^{2} \) |
| 31 | \( 1 - 230 T + p^{3} T^{2} \) |
| 37 | \( 1 - 54 T + p^{3} T^{2} \) |
| 41 | \( 1 + 195 T + p^{3} T^{2} \) |
| 43 | \( 1 - 300 T + p^{3} T^{2} \) |
| 47 | \( 1 - 480 T + p^{3} T^{2} \) |
| 53 | \( 1 + 322 T + p^{3} T^{2} \) |
| 59 | \( 1 + 560 T + p^{3} T^{2} \) |
| 61 | \( 1 + 730 T + p^{3} T^{2} \) |
| 67 | \( 1 + 255 T + p^{3} T^{2} \) |
| 71 | \( 1 - 40 T + p^{3} T^{2} \) |
| 73 | \( 1 + 317 T + p^{3} T^{2} \) |
| 79 | \( 1 - 830 T + p^{3} T^{2} \) |
| 83 | \( 1 + 75 T + p^{3} T^{2} \) |
| 89 | \( 1 + 705 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1434 T + p^{3} T^{2} \) |
| 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
−9.495029708811506740195456548311, −8.573058637563885270696784184215, −7.74122389417266865856741994405, −6.58990732172276525041354327964, −5.98323583451500850256674958195, −4.97784540411850012229358377703, −4.20054737058158072091501593653, −2.72323309106613696145344804648, −1.31498901398928734982893419191, 0,
1.31498901398928734982893419191, 2.72323309106613696145344804648, 4.20054737058158072091501593653, 4.97784540411850012229358377703, 5.98323583451500850256674958195, 6.58990732172276525041354327964, 7.74122389417266865856741994405, 8.573058637563885270696784184215, 9.495029708811506740195456548311
