| L(s) = 1 | + 0.273·2-s − 3-s − 1.92·4-s − 2.78·5-s − 0.273·6-s − 7-s − 1.07·8-s + 9-s − 0.761·10-s − 0.159·11-s + 1.92·12-s − 1.83·13-s − 0.273·14-s + 2.78·15-s + 3.55·16-s + 2.37·17-s + 0.273·18-s − 3.04·19-s + 5.36·20-s + 21-s − 0.0435·22-s − 5.95·23-s + 1.07·24-s + 2.76·25-s − 0.500·26-s − 27-s + 1.92·28-s + ⋯ |
| L(s) = 1 | + 0.193·2-s − 0.577·3-s − 0.962·4-s − 1.24·5-s − 0.111·6-s − 0.377·7-s − 0.379·8-s + 0.333·9-s − 0.240·10-s − 0.0480·11-s + 0.555·12-s − 0.507·13-s − 0.0730·14-s + 0.719·15-s + 0.889·16-s + 0.576·17-s + 0.0644·18-s − 0.697·19-s + 1.19·20-s + 0.218·21-s − 0.00928·22-s − 1.24·23-s + 0.218·24-s + 0.552·25-s − 0.0981·26-s − 0.192·27-s + 0.363·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.04504525685\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04504525685\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 - T \) |
| good | 2 | \( 1 - 0.273T + 2T^{2} \) |
| 5 | \( 1 + 2.78T + 5T^{2} \) |
| 11 | \( 1 + 0.159T + 11T^{2} \) |
| 13 | \( 1 + 1.83T + 13T^{2} \) |
| 17 | \( 1 - 2.37T + 17T^{2} \) |
| 19 | \( 1 + 3.04T + 19T^{2} \) |
| 23 | \( 1 + 5.95T + 23T^{2} \) |
| 29 | \( 1 + 6.31T + 29T^{2} \) |
| 31 | \( 1 + 0.338T + 31T^{2} \) |
| 37 | \( 1 - 5.86T + 37T^{2} \) |
| 41 | \( 1 - 4.58T + 41T^{2} \) |
| 43 | \( 1 + 12.4T + 43T^{2} \) |
| 47 | \( 1 + 12.6T + 47T^{2} \) |
| 53 | \( 1 + 5.02T + 53T^{2} \) |
| 59 | \( 1 + 7.53T + 59T^{2} \) |
| 61 | \( 1 - 7.97T + 61T^{2} \) |
| 67 | \( 1 + 13.6T + 67T^{2} \) |
| 71 | \( 1 + 5.42T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 + 4.15T + 79T^{2} \) |
| 83 | \( 1 + 13.5T + 83T^{2} \) |
| 89 | \( 1 + 6.26T + 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−7.82706202273629691193098570976, −7.29000463205599609976436898323, −6.28143730596671225598271225926, −5.74903169340699974233392448789, −4.85283616144880219215054034357, −4.32011088088912608044962105081, −3.71864162195081662757473782614, −2.99791678363600034905568744150, −1.56018218003362492140743287498, −0.10661380171370602823048727000,
0.10661380171370602823048727000, 1.56018218003362492140743287498, 2.99791678363600034905568744150, 3.71864162195081662757473782614, 4.32011088088912608044962105081, 4.85283616144880219215054034357, 5.74903169340699974233392448789, 6.28143730596671225598271225926, 7.29000463205599609976436898323, 7.82706202273629691193098570976
