| L(s) = 1 | − 1.99·2-s + 2.01·3-s + 1.98·4-s + 2.98·5-s − 4.01·6-s − 3.65·7-s + 0.0344·8-s + 1.05·9-s − 5.96·10-s + 11-s + 3.99·12-s − 6.84·13-s + 7.30·14-s + 6.01·15-s − 4.03·16-s + 17-s − 2.10·18-s − 3.07·19-s + 5.92·20-s − 7.36·21-s − 1.99·22-s − 5.38·23-s + 0.0693·24-s + 3.93·25-s + 13.6·26-s − 3.91·27-s − 7.25·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.16·3-s + 0.991·4-s + 1.33·5-s − 1.64·6-s − 1.38·7-s + 0.0121·8-s + 0.351·9-s − 1.88·10-s + 0.301·11-s + 1.15·12-s − 1.89·13-s + 1.95·14-s + 1.55·15-s − 1.00·16-s + 0.242·17-s − 0.496·18-s − 0.704·19-s + 1.32·20-s − 1.60·21-s − 0.425·22-s − 1.12·23-s + 0.0141·24-s + 0.786·25-s + 2.67·26-s − 0.753·27-s − 1.37·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.122262799\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.122262799\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 + 1.99T + 2T^{2} \) |
| 3 | \( 1 - 2.01T + 3T^{2} \) |
| 5 | \( 1 - 2.98T + 5T^{2} \) |
| 7 | \( 1 + 3.65T + 7T^{2} \) |
| 13 | \( 1 + 6.84T + 13T^{2} \) |
| 19 | \( 1 + 3.07T + 19T^{2} \) |
| 23 | \( 1 + 5.38T + 23T^{2} \) |
| 29 | \( 1 - 1.49T + 29T^{2} \) |
| 31 | \( 1 - 6.38T + 31T^{2} \) |
| 37 | \( 1 + 0.850T + 37T^{2} \) |
| 41 | \( 1 + 2.35T + 41T^{2} \) |
| 47 | \( 1 - 2.66T + 47T^{2} \) |
| 53 | \( 1 - 0.780T + 53T^{2} \) |
| 59 | \( 1 - 8.77T + 59T^{2} \) |
| 61 | \( 1 - 6.61T + 61T^{2} \) |
| 67 | \( 1 - 5.23T + 67T^{2} \) |
| 71 | \( 1 + 0.390T + 71T^{2} \) |
| 73 | \( 1 - 4.83T + 73T^{2} \) |
| 79 | \( 1 - 11.9T + 79T^{2} \) |
| 83 | \( 1 + 1.62T + 83T^{2} \) |
| 89 | \( 1 - 5.39T + 89T^{2} \) |
| 97 | \( 1 - 1.69T + 97T^{2} \) |
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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
−8.043736051563597050375030633763, −7.31772121994259961215127748522, −6.66098152550142800751259788470, −6.11022685486484438924775635543, −5.12637780971405965802779108595, −4.09713586673634859609767627625, −3.06802478266849762924015911893, −2.25840141579434046356476072171, −2.07467553228012739423950109585, −0.57536637179460433019986624972,
0.57536637179460433019986624972, 2.07467553228012739423950109585, 2.25840141579434046356476072171, 3.06802478266849762924015911893, 4.09713586673634859609767627625, 5.12637780971405965802779108595, 6.11022685486484438924775635543, 6.66098152550142800751259788470, 7.31772121994259961215127748522, 8.043736051563597050375030633763
