| L(s) = 1 | − 2.61·2-s − 2.05·3-s + 4.81·4-s − 0.541·5-s + 5.36·6-s − 3.52·7-s − 7.35·8-s + 1.21·9-s + 1.41·10-s + 1.19·11-s − 9.89·12-s − 4.15·13-s + 9.20·14-s + 1.11·15-s + 9.56·16-s + 17-s − 3.18·18-s − 2.49·19-s − 2.60·20-s + 7.23·21-s − 3.12·22-s − 1.46·23-s + 15.1·24-s − 4.70·25-s + 10.8·26-s + 3.65·27-s − 16.9·28-s + ⋯ |
| L(s) = 1 | − 1.84·2-s − 1.18·3-s + 2.40·4-s − 0.242·5-s + 2.18·6-s − 1.33·7-s − 2.60·8-s + 0.406·9-s + 0.446·10-s + 0.360·11-s − 2.85·12-s − 1.15·13-s + 2.45·14-s + 0.287·15-s + 2.39·16-s + 0.242·17-s − 0.750·18-s − 0.572·19-s − 0.583·20-s + 1.57·21-s − 0.666·22-s − 0.305·23-s + 3.08·24-s − 0.941·25-s + 2.12·26-s + 0.703·27-s − 3.20·28-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
| \(L(1)\) |
\(\approx\) |
\(0.1221348587\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1221348587\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.61T + 2T^{2} \) |
| 3 | \( 1 + 2.05T + 3T^{2} \) |
| 5 | \( 1 + 0.541T + 5T^{2} \) |
| 7 | \( 1 + 3.52T + 7T^{2} \) |
| 11 | \( 1 - 1.19T + 11T^{2} \) |
| 13 | \( 1 + 4.15T + 13T^{2} \) |
| 19 | \( 1 + 2.49T + 19T^{2} \) |
| 23 | \( 1 + 1.46T + 23T^{2} \) |
| 29 | \( 1 + 1.91T + 29T^{2} \) |
| 31 | \( 1 + 7.86T + 31T^{2} \) |
| 37 | \( 1 + 4.71T + 37T^{2} \) |
| 41 | \( 1 - 2.04T + 41T^{2} \) |
| 43 | \( 1 + 11.5T + 43T^{2} \) |
| 53 | \( 1 - 5.53T + 53T^{2} \) |
| 59 | \( 1 + 3.66T + 59T^{2} \) |
| 61 | \( 1 - 7.64T + 61T^{2} \) |
| 67 | \( 1 - 12.5T + 67T^{2} \) |
| 71 | \( 1 - 1.13T + 71T^{2} \) |
| 73 | \( 1 - 10.7T + 73T^{2} \) |
| 79 | \( 1 - 2.03T + 79T^{2} \) |
| 83 | \( 1 + 4.24T + 83T^{2} \) |
| 89 | \( 1 - 16.2T + 89T^{2} \) |
| 97 | \( 1 - 16.5T + 97T^{2} \) |
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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
−10.08123202175982708470154603779, −9.628256554279579474959042736884, −8.755793757607741652860280359684, −7.68745266612198273055623003117, −6.85525319151278952596735590323, −6.33481040909638494062704726752, −5.32800638700724651568886626407, −3.53858850921409134939056907925, −2.11553319229450587227928291408, −0.37138017184737382568281770802,
0.37138017184737382568281770802, 2.11553319229450587227928291408, 3.53858850921409134939056907925, 5.32800638700724651568886626407, 6.33481040909638494062704726752, 6.85525319151278952596735590323, 7.68745266612198273055623003117, 8.755793757607741652860280359684, 9.628256554279579474959042736884, 10.08123202175982708470154603779
