| L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s + 9-s + 5·16-s + 17-s − 2·18-s − 25-s − 6·32-s − 2·34-s + 3·36-s + 47-s + 49-s + 2·50-s + 2·53-s − 2·59-s + 7·64-s + 3·68-s − 4·72-s + 81-s + 2·83-s + 2·89-s − 2·94-s − 2·98-s − 3·100-s − 2·101-s − 2·103-s + ⋯ |
| L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s + 9-s + 5·16-s + 17-s − 2·18-s − 25-s − 6·32-s − 2·34-s + 3·36-s + 47-s + 49-s + 2·50-s + 2·53-s − 2·59-s + 7·64-s + 3·68-s − 4·72-s + 81-s + 2·83-s + 2·89-s − 2·94-s − 2·98-s − 3·100-s − 2·101-s − 2·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4586100528\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4586100528\) |
| \(L(1)\) |
|
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 | \( ( 1 + T )^{2} \) |
| 3 | \( ( 1 - T )( 1 + T ) \) |
| 5 | \( 1 + T^{2} \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )^{2} \) |
| 59 | \( ( 1 + T )^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )^{2} \) |
| 89 | \( ( 1 - T )^{2} \) |
| 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
−10.29820576207840850537714049679, −9.595906964192881264702390295351, −8.936588683823187090954747564502, −7.86870164786907834379830867003, −7.44783713661188941128173603628, −6.53198685684828429728004907793, −5.58468925574343620531674961962, −3.70806646406166267541434071666, −2.35894187413246931141314718096, −1.19290530175140966528755188267,
1.19290530175140966528755188267, 2.35894187413246931141314718096, 3.70806646406166267541434071666, 5.58468925574343620531674961962, 6.53198685684828429728004907793, 7.44783713661188941128173603628, 7.86870164786907834379830867003, 8.936588683823187090954747564502, 9.595906964192881264702390295351, 10.29820576207840850537714049679
