L(s) = 1 | + 2·3-s − 4·5-s + 4·7-s + 9-s + 2·11-s − 2·13-s − 8·15-s − 17-s + 8·19-s + 8·21-s + 8·23-s + 11·25-s − 4·27-s − 4·29-s + 4·31-s + 4·33-s − 16·35-s + 8·37-s − 4·39-s − 2·41-s − 4·43-s − 4·45-s + 9·49-s − 2·51-s + 6·53-s − 8·55-s + 16·57-s + ⋯ |
L(s) = 1 | + 1.15·3-s − 1.78·5-s + 1.51·7-s + 1/3·9-s + 0.603·11-s − 0.554·13-s − 2.06·15-s − 0.242·17-s + 1.83·19-s + 1.74·21-s + 1.66·23-s + 11/5·25-s − 0.769·27-s − 0.742·29-s + 0.718·31-s + 0.696·33-s − 2.70·35-s + 1.31·37-s − 0.640·39-s − 0.312·41-s − 0.609·43-s − 0.596·45-s + 9/7·49-s − 0.280·51-s + 0.824·53-s − 1.07·55-s + 2.11·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.133100331\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.133100331\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 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
−9.572965623577654595501127682823, −8.804450322213857736838695809493, −8.159865683869603549748339156235, −7.58593954838033377689898471873, −7.07867695292382400185019617775, −5.23037625113036685086022083735, −4.47425742597594277070011263195, −3.58917937326315127291596727489, −2.73268725075493987244108143967, −1.15983289619591015596664990780,
1.15983289619591015596664990780, 2.73268725075493987244108143967, 3.58917937326315127291596727489, 4.47425742597594277070011263195, 5.23037625113036685086022083735, 7.07867695292382400185019617775, 7.58593954838033377689898471873, 8.159865683869603549748339156235, 8.804450322213857736838695809493, 9.572965623577654595501127682823