| L(s) = 1 | − 2-s − 5-s + 2·7-s + 8-s + 9-s + 10-s − 13-s − 2·14-s − 16-s − 18-s − 23-s + 26-s + 2·29-s − 31-s − 2·35-s − 40-s − 43-s − 45-s + 46-s − 47-s + 3·49-s − 53-s + 2·56-s − 2·58-s − 61-s + 62-s + 2·63-s + ⋯ |
| L(s) = 1 | − 2-s − 5-s + 2·7-s + 8-s + 9-s + 10-s − 13-s − 2·14-s − 16-s − 18-s − 23-s + 26-s + 2·29-s − 31-s − 2·35-s − 40-s − 43-s − 45-s + 46-s − 47-s + 3·49-s − 53-s + 2·56-s − 2·58-s − 61-s + 62-s + 2·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4303185999\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4303185999\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 199 | \( 1 - T \) |
| good | 2 | \( 1 + T + T^{2} \) |
| 3 | \( ( 1 - T )( 1 + T ) \) |
| 5 | \( 1 + T + T^{2} \) |
| 7 | \( ( 1 - T )^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( ( 1 - T )^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( 1 + T + T^{2} \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−12.37400746033531004556361413159, −11.58075403325862385539087843620, −10.65177144352622755858934044383, −9.779264451042913963622914333402, −8.425719727080239166260616701156, −7.905416602642018013552333563786, −7.18387079154334138510864033589, −4.88681230623114224256968764865, −4.29247503558023135235147326245, −1.67161341998937103645071112925,
1.67161341998937103645071112925, 4.29247503558023135235147326245, 4.88681230623114224256968764865, 7.18387079154334138510864033589, 7.905416602642018013552333563786, 8.425719727080239166260616701156, 9.779264451042913963622914333402, 10.65177144352622755858934044383, 11.58075403325862385539087843620, 12.37400746033531004556361413159
