| L(s) = 1 | − 3-s + 4·5-s + 2·7-s + 9-s + 6·13-s − 4·15-s − 17-s + 4·19-s − 2·21-s − 6·23-s + 11·25-s − 27-s + 4·29-s + 6·31-s + 8·35-s + 4·37-s − 6·39-s − 10·41-s − 4·43-s + 4·45-s − 4·47-s − 3·49-s + 51-s + 2·53-s − 4·57-s + 12·59-s + 4·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.78·5-s + 0.755·7-s + 1/3·9-s + 1.66·13-s − 1.03·15-s − 0.242·17-s + 0.917·19-s − 0.436·21-s − 1.25·23-s + 11/5·25-s − 0.192·27-s + 0.742·29-s + 1.07·31-s + 1.35·35-s + 0.657·37-s − 0.960·39-s − 1.56·41-s − 0.609·43-s + 0.596·45-s − 0.583·47-s − 3/7·49-s + 0.140·51-s + 0.274·53-s − 0.529·57-s + 1.56·59-s + 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.754686586\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.754686586\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−8.574236112355339527046262755983, −8.127473857476079203645788388296, −6.79169938699478802588819175836, −6.32412925379975931990856588343, −5.63135877622582686929376756064, −5.08606107989467068869473300282, −4.11110200931759471218273933763, −2.89023851462989053245064085528, −1.75957782335756829082442787579, −1.17874391207982844132640722818,
1.17874391207982844132640722818, 1.75957782335756829082442787579, 2.89023851462989053245064085528, 4.11110200931759471218273933763, 5.08606107989467068869473300282, 5.63135877622582686929376756064, 6.32412925379975931990856588343, 6.79169938699478802588819175836, 8.127473857476079203645788388296, 8.574236112355339527046262755983
