| L(s) = 1 | − 2-s − 4-s − 2·5-s + 3·8-s + 2·10-s − 2·13-s − 16-s + 6·17-s + 4·19-s + 2·20-s − 25-s + 2·26-s − 2·29-s − 5·32-s − 6·34-s + 6·37-s − 4·38-s − 6·40-s − 2·41-s + 4·43-s + 50-s + 2·52-s − 6·53-s + 2·58-s + 12·59-s − 2·61-s + 7·64-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.894·5-s + 1.06·8-s + 0.632·10-s − 0.554·13-s − 1/4·16-s + 1.45·17-s + 0.917·19-s + 0.447·20-s − 1/5·25-s + 0.392·26-s − 0.371·29-s − 0.883·32-s − 1.02·34-s + 0.986·37-s − 0.648·38-s − 0.948·40-s − 0.312·41-s + 0.609·43-s + 0.141·50-s + 0.277·52-s − 0.824·53-s + 0.262·58-s + 1.56·59-s − 0.256·61-s + 7/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53361 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53361 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.005871172\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.005871172\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 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 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 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 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−14.43188836907869, −13.98397750440778, −13.50494424611324, −12.81174230351606, −12.34302680861143, −11.91400889792353, −11.28937863463768, −10.87847368449541, −10.10548715876596, −9.712060260288580, −9.422078382302612, −8.664295951197092, −8.029109736867478, −7.803068142106799, −7.340098297181168, −6.742656832568252, −5.729477011875389, −5.359597807999662, −4.670615164801450, −4.070936103233915, −3.531721572295379, −2.898735529886153, −1.917137616116670, −1.058485653896102, −0.4840284428379918,
0.4840284428379918, 1.058485653896102, 1.917137616116670, 2.898735529886153, 3.531721572295379, 4.070936103233915, 4.670615164801450, 5.359597807999662, 5.729477011875389, 6.742656832568252, 7.340098297181168, 7.803068142106799, 8.029109736867478, 8.664295951197092, 9.422078382302612, 9.712060260288580, 10.10548715876596, 10.87847368449541, 11.28937863463768, 11.91400889792353, 12.34302680861143, 12.81174230351606, 13.50494424611324, 13.98397750440778, 14.43188836907869
