| L(s) = 1 | − 2·3-s − 2·7-s + 9-s + 2·13-s + 6·17-s + 4·19-s + 4·21-s − 6·23-s + 4·27-s − 6·29-s − 4·31-s + 2·37-s − 4·39-s + 6·41-s − 10·43-s + 6·47-s − 3·49-s − 12·51-s − 6·53-s − 8·57-s − 12·59-s − 2·61-s − 2·63-s + 2·67-s + 12·69-s − 12·71-s − 2·73-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.755·7-s + 1/3·9-s + 0.554·13-s + 1.45·17-s + 0.917·19-s + 0.872·21-s − 1.25·23-s + 0.769·27-s − 1.11·29-s − 0.718·31-s + 0.328·37-s − 0.640·39-s + 0.937·41-s − 1.52·43-s + 0.875·47-s − 3/7·49-s − 1.68·51-s − 0.824·53-s − 1.05·57-s − 1.56·59-s − 0.256·61-s − 0.251·63-s + 0.244·67-s + 1.44·69-s − 1.42·71-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| 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
−9.258669147089765462577284951282, −8.059368180384024597996443374210, −7.35539571990596814095408613512, −6.29701258427927054860013270677, −5.82400824359360078327530562400, −5.13587744668360471627377345602, −3.88729748658685945494863770486, −3.05243989682609854421294881882, −1.38798538048163665478930118728, 0,
1.38798538048163665478930118728, 3.05243989682609854421294881882, 3.88729748658685945494863770486, 5.13587744668360471627377345602, 5.82400824359360078327530562400, 6.29701258427927054860013270677, 7.35539571990596814095408613512, 8.059368180384024597996443374210, 9.258669147089765462577284951282
