L(s) = 1 | + 3-s + 2·5-s + 4·7-s + 9-s − 11-s + 2·13-s + 2·15-s − 2·17-s + 4·21-s + 8·23-s − 25-s + 27-s + 6·29-s − 8·31-s − 33-s + 8·35-s − 6·37-s + 2·39-s − 2·41-s + 2·45-s + 8·47-s + 9·49-s − 2·51-s − 6·53-s − 2·55-s + 4·59-s − 6·61-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1.51·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s + 0.516·15-s − 0.485·17-s + 0.872·21-s + 1.66·23-s − 1/5·25-s + 0.192·27-s + 1.11·29-s − 1.43·31-s − 0.174·33-s + 1.35·35-s − 0.986·37-s + 0.320·39-s − 0.312·41-s + 0.298·45-s + 1.16·47-s + 9/7·49-s − 0.280·51-s − 0.824·53-s − 0.269·55-s + 0.520·59-s − 0.768·61-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
\(L(1)\) |
\(\approx\) |
\(3.170691475\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.170691475\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.815632295433106598675974855415, −8.598193922290110959822389745739, −7.57766902213530444274389318028, −6.91342543518502244816613622753, −5.79508255106269640718109448990, −5.08911955765620457569841389288, −4.32526881755629167390593741274, −3.10863384707893856832893049605, −2.07552052405763697992139591576, −1.33148151575350928602632304721,
1.33148151575350928602632304721, 2.07552052405763697992139591576, 3.10863384707893856832893049605, 4.32526881755629167390593741274, 5.08911955765620457569841389288, 5.79508255106269640718109448990, 6.91342543518502244816613622753, 7.57766902213530444274389318028, 8.598193922290110959822389745739, 8.815632295433106598675974855415