| L(s) = 1 | − 0.0537·2-s − 1.20·3-s − 1.99·4-s + 1.78·5-s + 0.0646·6-s + 2.05·7-s + 0.214·8-s − 1.55·9-s − 0.0957·10-s + 1.15·11-s + 2.40·12-s − 1.26·13-s − 0.110·14-s − 2.14·15-s + 3.98·16-s − 17-s + 0.0833·18-s − 0.107·19-s − 3.56·20-s − 2.46·21-s − 0.0622·22-s − 1.11·23-s − 0.258·24-s − 1.81·25-s + 0.0676·26-s + 5.47·27-s − 4.09·28-s + ⋯ |
| L(s) = 1 | − 0.0379·2-s − 0.694·3-s − 0.998·4-s + 0.797·5-s + 0.0263·6-s + 0.775·7-s + 0.0758·8-s − 0.517·9-s − 0.0302·10-s + 0.349·11-s + 0.693·12-s − 0.349·13-s − 0.0294·14-s − 0.553·15-s + 0.995·16-s − 0.242·17-s + 0.0196·18-s − 0.0247·19-s − 0.796·20-s − 0.538·21-s − 0.0132·22-s − 0.233·23-s − 0.0527·24-s − 0.363·25-s + 0.0132·26-s + 1.05·27-s − 0.774·28-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
| \(L(1)\) |
\(\approx\) |
\(1.099424484\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.099424484\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.0537T + 2T^{2} \) |
| 3 | \( 1 + 1.20T + 3T^{2} \) |
| 5 | \( 1 - 1.78T + 5T^{2} \) |
| 7 | \( 1 - 2.05T + 7T^{2} \) |
| 11 | \( 1 - 1.15T + 11T^{2} \) |
| 13 | \( 1 + 1.26T + 13T^{2} \) |
| 19 | \( 1 + 0.107T + 19T^{2} \) |
| 23 | \( 1 + 1.11T + 23T^{2} \) |
| 29 | \( 1 - 7.49T + 29T^{2} \) |
| 31 | \( 1 - 7.10T + 31T^{2} \) |
| 37 | \( 1 - 1.62T + 37T^{2} \) |
| 41 | \( 1 - 4.79T + 41T^{2} \) |
| 43 | \( 1 - 9.71T + 43T^{2} \) |
| 53 | \( 1 - 8.88T + 53T^{2} \) |
| 59 | \( 1 + 7.34T + 59T^{2} \) |
| 61 | \( 1 - 14.3T + 61T^{2} \) |
| 67 | \( 1 + 4.99T + 67T^{2} \) |
| 71 | \( 1 - 7.36T + 71T^{2} \) |
| 73 | \( 1 - 16.7T + 73T^{2} \) |
| 79 | \( 1 + 3.88T + 79T^{2} \) |
| 83 | \( 1 - 0.850T + 83T^{2} \) |
| 89 | \( 1 + 8.65T + 89T^{2} \) |
| 97 | \( 1 - 6.75T + 97T^{2} \) |
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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
−10.20713874482599956309970380411, −9.485606803990180934964428758491, −8.609459111435258024513275832568, −7.916589832836109038584511335163, −6.53650806425601838832223534023, −5.72480642714255841086601083873, −4.98783815935956889440578380261, −4.16860335946389408324468678140, −2.53335608244172791228644232923, −0.932484863996735032562323607441,
0.932484863996735032562323607441, 2.53335608244172791228644232923, 4.16860335946389408324468678140, 4.98783815935956889440578380261, 5.72480642714255841086601083873, 6.53650806425601838832223534023, 7.916589832836109038584511335163, 8.609459111435258024513275832568, 9.485606803990180934964428758491, 10.20713874482599956309970380411
