| L(s) = 1 | − 1.25·2-s − 0.414·4-s − 1.25·5-s + 3.04·8-s + 1.58·10-s − 4.82·11-s + 1.17·13-s − 3·16-s − 4.29·17-s + 1.58·19-s + 0.521·20-s + 6.07·22-s + 6.60·23-s − 3.41·25-s − 1.47·26-s − 5.03·29-s + 10.6·31-s − 2.30·32-s + 5.41·34-s − 5.24·37-s − 1.99·38-s − 3.82·40-s − 0.521·41-s + 7.07·43-s + 1.99·44-s − 8.31·46-s − 11.4·47-s + ⋯ |
| L(s) = 1 | − 0.890·2-s − 0.207·4-s − 0.563·5-s + 1.07·8-s + 0.501·10-s − 1.45·11-s + 0.324·13-s − 0.750·16-s − 1.04·17-s + 0.363·19-s + 0.116·20-s + 1.29·22-s + 1.37·23-s − 0.682·25-s − 0.289·26-s − 0.935·29-s + 1.91·31-s − 0.407·32-s + 0.928·34-s − 0.861·37-s − 0.323·38-s − 0.605·40-s − 0.0814·41-s + 1.07·43-s + 0.301·44-s − 1.22·46-s − 1.66·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5864844480\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5864844480\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 1.25T + 2T^{2} \) |
| 5 | \( 1 + 1.25T + 5T^{2} \) |
| 11 | \( 1 + 4.82T + 11T^{2} \) |
| 13 | \( 1 - 1.17T + 13T^{2} \) |
| 17 | \( 1 + 4.29T + 17T^{2} \) |
| 19 | \( 1 - 1.58T + 19T^{2} \) |
| 23 | \( 1 - 6.60T + 23T^{2} \) |
| 29 | \( 1 + 5.03T + 29T^{2} \) |
| 31 | \( 1 - 10.6T + 31T^{2} \) |
| 37 | \( 1 + 5.24T + 37T^{2} \) |
| 41 | \( 1 + 0.521T + 41T^{2} \) |
| 43 | \( 1 - 7.07T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 + 1.04T + 53T^{2} \) |
| 59 | \( 1 - 1.78T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 - 5.65T + 67T^{2} \) |
| 71 | \( 1 - 8.81T + 71T^{2} \) |
| 73 | \( 1 - 8.58T + 73T^{2} \) |
| 79 | \( 1 + 13.8T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 - 17.7T + 89T^{2} \) |
| 97 | \( 1 - 9.65T + 97T^{2} \) |
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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
−9.575477029136657268361725857056, −8.744455109643777300458549400916, −8.098194586219279832560871411025, −7.51717018478858746730581983365, −6.60246917122514352707098332572, −5.26374977711413510701342090043, −4.61966293064873693599803954730, −3.47935950519704869468560408985, −2.20475677507131938368984025300, −0.62841134942850118325255961102,
0.62841134942850118325255961102, 2.20475677507131938368984025300, 3.47935950519704869468560408985, 4.61966293064873693599803954730, 5.26374977711413510701342090043, 6.60246917122514352707098332572, 7.51717018478858746730581983365, 8.098194586219279832560871411025, 8.744455109643777300458549400916, 9.575477029136657268361725857056
