L(s) = 1 | − 2·4-s + 3·5-s + 7-s + 13-s + 4·16-s + 6·17-s − 7·19-s − 6·20-s − 3·23-s + 4·25-s − 2·28-s + 9·29-s + 5·31-s + 3·35-s + 2·37-s + 6·41-s − 43-s − 3·47-s + 49-s − 2·52-s + 9·53-s − 10·61-s − 8·64-s + 3·65-s + 14·67-s − 12·68-s + 6·71-s + ⋯ |
L(s) = 1 | − 4-s + 1.34·5-s + 0.377·7-s + 0.277·13-s + 16-s + 1.45·17-s − 1.60·19-s − 1.34·20-s − 0.625·23-s + 4/5·25-s − 0.377·28-s + 1.67·29-s + 0.898·31-s + 0.507·35-s + 0.328·37-s + 0.937·41-s − 0.152·43-s − 0.437·47-s + 1/7·49-s − 0.277·52-s + 1.23·53-s − 1.28·61-s − 64-s + 0.372·65-s + 1.71·67-s − 1.45·68-s + 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.674785814\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.674785814\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−10.07601453210756625703688021182, −9.542604600630706787025977559837, −8.524079546582096676065943493512, −8.011832969610721686895020035599, −6.49992505927329333857395184343, −5.79128008848021850482104511614, −4.94336211753658685120988545492, −3.97318584788109067523740792933, −2.53593630462827092595495233291, −1.17094069297371334537206633547,
1.17094069297371334537206633547, 2.53593630462827092595495233291, 3.97318584788109067523740792933, 4.94336211753658685120988545492, 5.79128008848021850482104511614, 6.49992505927329333857395184343, 8.011832969610721686895020035599, 8.524079546582096676065943493512, 9.542604600630706787025977559837, 10.07601453210756625703688021182