| L(s) = 1 | − 0.574·2-s + 1.45·3-s − 1.66·4-s − 2.31·5-s − 0.837·6-s − 3.08·7-s + 2.10·8-s − 0.879·9-s + 1.33·10-s + 11-s − 2.43·12-s − 0.647·13-s + 1.77·14-s − 3.37·15-s + 2.12·16-s + 5.50·17-s + 0.505·18-s − 2.59·19-s + 3.86·20-s − 4.48·21-s − 0.574·22-s − 3.99·23-s + 3.07·24-s + 0.367·25-s + 0.372·26-s − 5.64·27-s + 5.14·28-s + ⋯ |
| L(s) = 1 | − 0.406·2-s + 0.840·3-s − 0.834·4-s − 1.03·5-s − 0.341·6-s − 1.16·7-s + 0.745·8-s − 0.293·9-s + 0.421·10-s + 0.301·11-s − 0.701·12-s − 0.179·13-s + 0.473·14-s − 0.871·15-s + 0.531·16-s + 1.33·17-s + 0.119·18-s − 0.594·19-s + 0.864·20-s − 0.979·21-s − 0.122·22-s − 0.832·23-s + 0.627·24-s + 0.0734·25-s + 0.0730·26-s − 1.08·27-s + 0.972·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8022791181\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8022791181\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 131 | \( 1 + T \) |
| good | 2 | \( 1 + 0.574T + 2T^{2} \) |
| 3 | \( 1 - 1.45T + 3T^{2} \) |
| 5 | \( 1 + 2.31T + 5T^{2} \) |
| 7 | \( 1 + 3.08T + 7T^{2} \) |
| 13 | \( 1 + 0.647T + 13T^{2} \) |
| 17 | \( 1 - 5.50T + 17T^{2} \) |
| 19 | \( 1 + 2.59T + 19T^{2} \) |
| 23 | \( 1 + 3.99T + 23T^{2} \) |
| 29 | \( 1 + 1.55T + 29T^{2} \) |
| 31 | \( 1 - 6.06T + 31T^{2} \) |
| 37 | \( 1 - 9.24T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 - 9.39T + 43T^{2} \) |
| 47 | \( 1 - 4.79T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 + 5.95T + 59T^{2} \) |
| 61 | \( 1 - 11.3T + 61T^{2} \) |
| 67 | \( 1 - 11.8T + 67T^{2} \) |
| 71 | \( 1 - 14.0T + 71T^{2} \) |
| 73 | \( 1 - 1.41T + 73T^{2} \) |
| 79 | \( 1 - 14.2T + 79T^{2} \) |
| 83 | \( 1 - 6.00T + 83T^{2} \) |
| 89 | \( 1 - 4.01T + 89T^{2} \) |
| 97 | \( 1 + 16.7T + 97T^{2} \) |
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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
−9.637828218432350470420706441711, −8.609784862130120309278468499309, −8.089695653885960726830786337051, −7.51075326094104741948229884569, −6.36453319205444275184351608764, −5.33670381852694557323843128861, −3.93885744705375536773501594917, −3.71515483697268958450587191491, −2.54712843871629404068301870797, −0.64168996377230418531193097606,
0.64168996377230418531193097606, 2.54712843871629404068301870797, 3.71515483697268958450587191491, 3.93885744705375536773501594917, 5.33670381852694557323843128861, 6.36453319205444275184351608764, 7.51075326094104741948229884569, 8.089695653885960726830786337051, 8.609784862130120309278468499309, 9.637828218432350470420706441711
