| L(s) = 1 | − 3-s − 4-s + 7-s + 9-s + 12-s + 2·13-s + 16-s − 19-s − 21-s − 25-s − 27-s − 28-s + 2·31-s − 36-s − 2·39-s − 2·43-s − 48-s + 49-s − 2·52-s + 57-s + 63-s − 64-s + 75-s + 76-s + 81-s + 84-s + 2·91-s + ⋯ |
| L(s) = 1 | − 3-s − 4-s + 7-s + 9-s + 12-s + 2·13-s + 16-s − 19-s − 21-s − 25-s − 27-s − 28-s + 2·31-s − 36-s − 2·39-s − 2·43-s − 48-s + 49-s − 2·52-s + 57-s + 63-s − 64-s + 75-s + 76-s + 81-s + 84-s + 2·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5941442176\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5941442176\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + T^{2} \) |
| 5 | \( 1 + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( ( 1 - T )^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 + T )^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 + T )^{2} \) |
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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
−11.43531789730544054212942034762, −10.69031857678941998590533755169, −9.852184327363610117644180104244, −8.565301410105063599651733364189, −8.079768711600176609151322313087, −6.52346481263202133528329542436, −5.68083507792614620484849484540, −4.65098871184700684507876309771, −3.87052350923602871652968544839, −1.37167694822665074547560374203,
1.37167694822665074547560374203, 3.87052350923602871652968544839, 4.65098871184700684507876309771, 5.68083507792614620484849484540, 6.52346481263202133528329542436, 8.079768711600176609151322313087, 8.565301410105063599651733364189, 9.852184327363610117644180104244, 10.69031857678941998590533755169, 11.43531789730544054212942034762
