| L(s) = 1 | + 3-s − 4·7-s + 9-s − 13-s − 2·17-s + 8·19-s − 4·21-s − 4·23-s + 27-s + 6·29-s − 8·31-s − 6·37-s − 39-s − 6·41-s + 4·43-s − 4·47-s + 9·49-s − 2·51-s − 6·53-s + 8·57-s + 8·59-s − 10·61-s − 4·63-s − 4·67-s − 4·69-s + 8·71-s + 14·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.51·7-s + 1/3·9-s − 0.277·13-s − 0.485·17-s + 1.83·19-s − 0.872·21-s − 0.834·23-s + 0.192·27-s + 1.11·29-s − 1.43·31-s − 0.986·37-s − 0.160·39-s − 0.937·41-s + 0.609·43-s − 0.583·47-s + 9/7·49-s − 0.280·51-s − 0.824·53-s + 1.05·57-s + 1.04·59-s − 1.28·61-s − 0.503·63-s − 0.488·67-s − 0.481·69-s + 0.949·71-s + 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.564187452\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.564187452\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−15.32610863756210, −14.30907387701451, −14.07553244094186, −13.56770755024404, −12.98653588339760, −12.50887565265767, −12.04961564460841, −11.44359873974562, −10.66751686248170, −10.07138174506406, −9.676546220166185, −9.255032215941829, −8.681505286106220, −7.973974684521878, −7.365656482724132, −6.852774551367829, −6.361562245547142, −5.598965039858184, −5.034950335464455, −4.174451249036329, −3.401266885618667, −3.191247828471999, −2.368056946509386, −1.539414231042843, −0.4515744376910160,
0.4515744376910160, 1.539414231042843, 2.368056946509386, 3.191247828471999, 3.401266885618667, 4.174451249036329, 5.034950335464455, 5.598965039858184, 6.361562245547142, 6.852774551367829, 7.365656482724132, 7.973974684521878, 8.681505286106220, 9.255032215941829, 9.676546220166185, 10.07138174506406, 10.66751686248170, 11.44359873974562, 12.04961564460841, 12.50887565265767, 12.98653588339760, 13.56770755024404, 14.07553244094186, 14.30907387701451, 15.32610863756210
