| L(s) = 1 | − 2-s + 2·5-s − 7-s + 8-s + 9-s − 2·10-s − 11-s − 13-s + 14-s − 16-s − 18-s + 2·19-s + 22-s + 3·25-s + 26-s − 29-s − 2·35-s − 2·38-s + 2·40-s + 2·45-s − 3·50-s − 53-s − 2·55-s − 56-s + 58-s − 61-s − 63-s + ⋯ |
| L(s) = 1 | − 2-s + 2·5-s − 7-s + 8-s + 9-s − 2·10-s − 11-s − 13-s + 14-s − 16-s − 18-s + 2·19-s + 22-s + 3·25-s + 26-s − 29-s − 2·35-s − 2·38-s + 2·40-s + 2·45-s − 3·50-s − 53-s − 2·55-s − 56-s + 58-s − 61-s − 63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 439 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 439 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6170397827\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6170397827\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 439 | \( 1 - T \) |
| good | 2 | \( 1 + T + T^{2} \) |
| 3 | \( ( 1 - T )( 1 + T ) \) |
| 5 | \( ( 1 - T )^{2} \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( 1 + T + T^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 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
−10.75649899884994662595279996459, −9.900396697318622798670496097307, −9.760371212740301129454149866057, −9.134397475148824262538315274588, −7.63046914757755305323007828898, −6.93100875928359441996626678638, −5.67777756447323515866922714090, −4.86821437601776639546880056479, −2.87253281375891967462898987716, −1.56772093530706485978459515838,
1.56772093530706485978459515838, 2.87253281375891967462898987716, 4.86821437601776639546880056479, 5.67777756447323515866922714090, 6.93100875928359441996626678638, 7.63046914757755305323007828898, 9.134397475148824262538315274588, 9.760371212740301129454149866057, 9.900396697318622798670496097307, 10.75649899884994662595279996459
