L(s) = 1 | − 3-s + 2·5-s + 9-s − 2·13-s − 2·15-s − 6·17-s + 4·19-s + 4·23-s − 25-s − 27-s − 6·29-s − 8·31-s + 10·37-s + 2·39-s + 10·41-s + 12·43-s + 2·45-s − 8·47-s + 6·51-s − 6·53-s − 4·57-s − 4·59-s − 10·61-s − 4·65-s + 12·67-s − 4·69-s − 4·71-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1/3·9-s − 0.554·13-s − 0.516·15-s − 1.45·17-s + 0.917·19-s + 0.834·23-s − 1/5·25-s − 0.192·27-s − 1.11·29-s − 1.43·31-s + 1.64·37-s + 0.320·39-s + 1.56·41-s + 1.82·43-s + 0.298·45-s − 1.16·47-s + 0.840·51-s − 0.824·53-s − 0.529·57-s − 0.520·59-s − 1.28·61-s − 0.496·65-s + 1.46·67-s − 0.481·69-s − 0.474·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 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 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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
−7.37858363838248446160078701342, −6.57789868247435997998295651334, −5.91049326221092939323680020702, −5.45341968887645466689109288381, −4.65714616628027431432206560326, −4.00965934400221477305823637249, −2.85571758992652827608417970940, −2.15150344614261598632778619468, −1.25423182140621920036242037665, 0,
1.25423182140621920036242037665, 2.15150344614261598632778619468, 2.85571758992652827608417970940, 4.00965934400221477305823637249, 4.65714616628027431432206560326, 5.45341968887645466689109288381, 5.91049326221092939323680020702, 6.57789868247435997998295651334, 7.37858363838248446160078701342