| L(s) = 1 | + 3-s + 4·7-s + 9-s − 6·13-s + 6·17-s + 4·19-s + 4·21-s + 27-s + 2·29-s − 8·31-s − 10·37-s − 6·39-s − 6·41-s − 43-s + 8·47-s + 9·49-s + 6·51-s + 6·53-s + 4·57-s + 10·61-s + 4·63-s − 4·67-s − 8·71-s + 6·73-s + 81-s + 16·83-s + 2·87-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.51·7-s + 1/3·9-s − 1.66·13-s + 1.45·17-s + 0.917·19-s + 0.872·21-s + 0.192·27-s + 0.371·29-s − 1.43·31-s − 1.64·37-s − 0.960·39-s − 0.937·41-s − 0.152·43-s + 1.16·47-s + 9/7·49-s + 0.840·51-s + 0.824·53-s + 0.529·57-s + 1.28·61-s + 0.503·63-s − 0.488·67-s − 0.949·71-s + 0.702·73-s + 1/9·81-s + 1.75·83-s + 0.214·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.782757516\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.782757516\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 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
−14.54665534800345, −14.05139295577399, −13.72536045034262, −12.96232230463950, −12.20774177011218, −12.00502622684652, −11.64456738035870, −10.63972624341195, −10.48452219300279, −9.751495680713697, −9.324492122024027, −8.675180896059776, −8.122866670092710, −7.637938333028000, −7.281548604756946, −6.801728580033069, −5.538527270662368, −5.324146452136019, −4.888085651964377, −4.088856894814249, −3.461959020695263, −2.789391670000864, −2.024051158375667, −1.560410552816762, −0.6590617167263241,
0.6590617167263241, 1.560410552816762, 2.024051158375667, 2.789391670000864, 3.461959020695263, 4.088856894814249, 4.888085651964377, 5.324146452136019, 5.538527270662368, 6.801728580033069, 7.281548604756946, 7.637938333028000, 8.122866670092710, 8.675180896059776, 9.324492122024027, 9.751495680713697, 10.48452219300279, 10.63972624341195, 11.64456738035870, 12.00502622684652, 12.20774177011218, 12.96232230463950, 13.72536045034262, 14.05139295577399, 14.54665534800345
