| L(s) = 1 | + 1.53·2-s − 1.07·3-s + 0.362·4-s − 2.28·5-s − 1.65·6-s + 2.77·7-s − 2.51·8-s − 1.84·9-s − 3.51·10-s + 5.21·11-s − 0.389·12-s + 4.60·13-s + 4.26·14-s + 2.45·15-s − 4.59·16-s − 17-s − 2.83·18-s + 5.93·19-s − 0.828·20-s − 2.98·21-s + 8.01·22-s + 8.91·23-s + 2.70·24-s + 0.218·25-s + 7.07·26-s + 5.20·27-s + 1.00·28-s + ⋯ |
| L(s) = 1 | + 1.08·2-s − 0.620·3-s + 0.181·4-s − 1.02·5-s − 0.674·6-s + 1.04·7-s − 0.889·8-s − 0.615·9-s − 1.11·10-s + 1.57·11-s − 0.112·12-s + 1.27·13-s + 1.13·14-s + 0.633·15-s − 1.14·16-s − 0.242·17-s − 0.668·18-s + 1.36·19-s − 0.185·20-s − 0.650·21-s + 1.70·22-s + 1.85·23-s + 0.551·24-s + 0.0436·25-s + 1.38·26-s + 1.00·27-s + 0.190·28-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
| \(L(1)\) |
\(\approx\) |
\(1.927611146\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.927611146\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.53T + 2T^{2} \) |
| 3 | \( 1 + 1.07T + 3T^{2} \) |
| 5 | \( 1 + 2.28T + 5T^{2} \) |
| 7 | \( 1 - 2.77T + 7T^{2} \) |
| 11 | \( 1 - 5.21T + 11T^{2} \) |
| 13 | \( 1 - 4.60T + 13T^{2} \) |
| 19 | \( 1 - 5.93T + 19T^{2} \) |
| 23 | \( 1 - 8.91T + 23T^{2} \) |
| 29 | \( 1 - 0.612T + 29T^{2} \) |
| 31 | \( 1 + 2.06T + 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 - 3.07T + 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 53 | \( 1 - 3.62T + 53T^{2} \) |
| 59 | \( 1 - 3.88T + 59T^{2} \) |
| 61 | \( 1 + 12.7T + 61T^{2} \) |
| 67 | \( 1 + 6.68T + 67T^{2} \) |
| 71 | \( 1 - 6.43T + 71T^{2} \) |
| 73 | \( 1 - 2.21T + 73T^{2} \) |
| 79 | \( 1 + 13.7T + 79T^{2} \) |
| 83 | \( 1 - 3.13T + 83T^{2} \) |
| 89 | \( 1 - 16.8T + 89T^{2} \) |
| 97 | \( 1 + 7.73T + 97T^{2} \) |
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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
−10.88974566848455879294567536478, −9.018573770465422100832198779134, −8.787303233255958490363732872802, −7.51867060606043058345547970847, −6.53516387852656885579459198230, −5.62520699091021350949536198692, −4.85257079495003326986335225255, −3.95903390187270845800664027573, −3.22839084282675288007803878221, −1.08693802752278158186583907909,
1.08693802752278158186583907909, 3.22839084282675288007803878221, 3.95903390187270845800664027573, 4.85257079495003326986335225255, 5.62520699091021350949536198692, 6.53516387852656885579459198230, 7.51867060606043058345547970847, 8.787303233255958490363732872802, 9.018573770465422100832198779134, 10.88974566848455879294567536478
