| L(s) = 1 | − 2-s + 2·3-s − 4-s + 4·5-s − 2·6-s − 2·7-s + 3·8-s + 9-s − 4·10-s − 2·12-s + 2·13-s + 2·14-s + 8·15-s − 16-s + 17-s − 18-s + 4·19-s − 4·20-s − 4·21-s − 4·23-s + 6·24-s + 11·25-s − 2·26-s − 4·27-s + 2·28-s + 8·29-s − 8·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s − 1/2·4-s + 1.78·5-s − 0.816·6-s − 0.755·7-s + 1.06·8-s + 1/3·9-s − 1.26·10-s − 0.577·12-s + 0.554·13-s + 0.534·14-s + 2.06·15-s − 1/4·16-s + 0.242·17-s − 0.235·18-s + 0.917·19-s − 0.894·20-s − 0.872·21-s − 0.834·23-s + 1.22·24-s + 11/5·25-s − 0.392·26-s − 0.769·27-s + 0.377·28-s + 1.48·29-s − 1.46·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.782342728\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.782342728\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 - T \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−9.923805233114826705426209355465, −9.510500167185474756907047741981, −8.659648938238949671437076516991, −8.185517248538253094842685346692, −6.87454595313213151825457041697, −5.97785207334378606624688271601, −4.95558297373443957549295237784, −3.49977739762465808068312303580, −2.51032747149775727153897197062, −1.32013274406742367289000975025,
1.32013274406742367289000975025, 2.51032747149775727153897197062, 3.49977739762465808068312303580, 4.95558297373443957549295237784, 5.97785207334378606624688271601, 6.87454595313213151825457041697, 8.185517248538253094842685346692, 8.659648938238949671437076516991, 9.510500167185474756907047741981, 9.923805233114826705426209355465
