| L(s) = 1 | − 3-s − 2·7-s + 9-s + 2·13-s − 17-s + 4·19-s + 2·21-s + 6·23-s − 5·25-s − 27-s + 10·31-s + 8·37-s − 2·39-s + 6·41-s + 4·43-s − 12·47-s − 3·49-s + 51-s + 6·53-s − 4·57-s + 12·59-s + 8·61-s − 2·63-s + 4·67-s − 6·69-s − 6·71-s + 2·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s + 0.554·13-s − 0.242·17-s + 0.917·19-s + 0.436·21-s + 1.25·23-s − 25-s − 0.192·27-s + 1.79·31-s + 1.31·37-s − 0.320·39-s + 0.937·41-s + 0.609·43-s − 1.75·47-s − 3/7·49-s + 0.140·51-s + 0.824·53-s − 0.529·57-s + 1.56·59-s + 1.02·61-s − 0.251·63-s + 0.488·67-s − 0.722·69-s − 0.712·71-s + 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.194137070\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.194137070\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 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 | \( 1 + 18 T + p T^{2} \) |
| 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
−10.08694532952722620244587438047, −9.629005176149752438633105274587, −8.584868551115419667042947592500, −7.57323216500839527256914149425, −6.62762171992257665130856519803, −5.95967651264422152670977788910, −4.94376122657450226002634205632, −3.84609690192901140186989283204, −2.71372337045158087575655535987, −0.943136411131521382953657369703,
0.943136411131521382953657369703, 2.71372337045158087575655535987, 3.84609690192901140186989283204, 4.94376122657450226002634205632, 5.95967651264422152670977788910, 6.62762171992257665130856519803, 7.57323216500839527256914149425, 8.584868551115419667042947592500, 9.629005176149752438633105274587, 10.08694532952722620244587438047
