| L(s) = 1 | − 2.66·2-s + 5.12·4-s + 1.45·5-s + 7-s − 8.33·8-s − 3.88·10-s + 1.54·11-s + 5.88·13-s − 2.66·14-s + 12.0·16-s + 6.79·17-s − 6.24·19-s + 7.45·20-s − 4.12·22-s − 2.90·23-s − 2.88·25-s − 15.7·26-s + 5.12·28-s − 3.88·29-s + 2·31-s − 15.3·32-s − 18.1·34-s + 1.45·35-s + 5·37-s + 16.6·38-s − 12.1·40-s + 2.24·41-s + ⋯ |
| L(s) = 1 | − 1.88·2-s + 2.56·4-s + 0.650·5-s + 0.377·7-s − 2.94·8-s − 1.22·10-s + 0.465·11-s + 1.63·13-s − 0.713·14-s + 3.00·16-s + 1.64·17-s − 1.43·19-s + 1.66·20-s − 0.879·22-s − 0.606·23-s − 0.576·25-s − 3.07·26-s + 0.968·28-s − 0.721·29-s + 0.359·31-s − 2.71·32-s − 3.10·34-s + 0.245·35-s + 0.821·37-s + 2.70·38-s − 1.91·40-s + 0.351·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8080898510\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8080898510\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 2 | \( 1 + 2.66T + 2T^{2} \) |
| 5 | \( 1 - 1.45T + 5T^{2} \) |
| 11 | \( 1 - 1.54T + 11T^{2} \) |
| 13 | \( 1 - 5.88T + 13T^{2} \) |
| 17 | \( 1 - 6.79T + 17T^{2} \) |
| 19 | \( 1 + 6.24T + 19T^{2} \) |
| 23 | \( 1 + 2.90T + 23T^{2} \) |
| 29 | \( 1 + 3.88T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 5T + 37T^{2} \) |
| 41 | \( 1 - 2.24T + 41T^{2} \) |
| 43 | \( 1 + 7.13T + 43T^{2} \) |
| 47 | \( 1 - 5.33T + 47T^{2} \) |
| 53 | \( 1 - 9.79T + 53T^{2} \) |
| 59 | \( 1 - 4.67T + 59T^{2} \) |
| 61 | \( 1 + 2.36T + 61T^{2} \) |
| 67 | \( 1 - 3.36T + 67T^{2} \) |
| 71 | \( 1 + 1.36T + 71T^{2} \) |
| 73 | \( 1 + 1.88T + 73T^{2} \) |
| 79 | \( 1 - 3.36T + 79T^{2} \) |
| 83 | \( 1 + 2.24T + 83T^{2} \) |
| 89 | \( 1 + 0.793T + 89T^{2} \) |
| 97 | \( 1 - 10.2T + 97T^{2} \) |
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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
−10.42659999134972895711053023129, −9.880975183866917534673380678283, −8.919123843560965511580620413877, −8.327251558468038827133760404133, −7.51875064920561374435167811396, −6.32017146365063704166652220210, −5.82174409256645307220823685423, −3.71313678793279702236690079027, −2.13978568383100041861733523018, −1.14553341242214246398375788316,
1.14553341242214246398375788316, 2.13978568383100041861733523018, 3.71313678793279702236690079027, 5.82174409256645307220823685423, 6.32017146365063704166652220210, 7.51875064920561374435167811396, 8.327251558468038827133760404133, 8.919123843560965511580620413877, 9.880975183866917534673380678283, 10.42659999134972895711053023129
