| L(s) = 1 | + 3-s − 4·5-s + 2·7-s + 9-s − 11-s − 4·15-s − 6·17-s + 4·19-s + 2·21-s + 6·23-s + 11·25-s + 27-s − 6·29-s − 33-s − 8·35-s − 6·37-s − 10·41-s − 8·43-s − 4·45-s − 6·47-s − 3·49-s − 6·51-s + 12·53-s + 4·55-s + 4·57-s − 8·59-s − 4·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.78·5-s + 0.755·7-s + 1/3·9-s − 0.301·11-s − 1.03·15-s − 1.45·17-s + 0.917·19-s + 0.436·21-s + 1.25·23-s + 11/5·25-s + 0.192·27-s − 1.11·29-s − 0.174·33-s − 1.35·35-s − 0.986·37-s − 1.56·41-s − 1.21·43-s − 0.596·45-s − 0.875·47-s − 3/7·49-s − 0.840·51-s + 1.64·53-s + 0.539·55-s + 0.529·57-s − 1.04·59-s − 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 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 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.634825016109692859963514974092, −7.948856795491039403583542835581, −7.31799682783557799784216934179, −6.75143889548196905356209799291, −5.13616906335360273416256902725, −4.62172996771965878028685248790, −3.67718863281571613929216971003, −2.99465322081378559589642172143, −1.60371736211155010008787601511, 0,
1.60371736211155010008787601511, 2.99465322081378559589642172143, 3.67718863281571613929216971003, 4.62172996771965878028685248790, 5.13616906335360273416256902725, 6.75143889548196905356209799291, 7.31799682783557799784216934179, 7.948856795491039403583542835581, 8.634825016109692859963514974092
