| L(s) = 1 | − 1.71·2-s − 2.58·3-s + 0.937·4-s + 0.849·5-s + 4.43·6-s − 2.18·7-s + 1.82·8-s + 3.68·9-s − 1.45·10-s − 6.43·11-s − 2.42·12-s − 0.0958·13-s + 3.75·14-s − 2.19·15-s − 4.99·16-s + 1.27·17-s − 6.31·18-s − 2.66·19-s + 0.795·20-s + 5.66·21-s + 11.0·22-s − 3.64·23-s − 4.70·24-s − 4.27·25-s + 0.164·26-s − 1.77·27-s − 2.05·28-s + ⋯ |
| L(s) = 1 | − 1.21·2-s − 1.49·3-s + 0.468·4-s + 0.379·5-s + 1.80·6-s − 0.827·7-s + 0.643·8-s + 1.22·9-s − 0.460·10-s − 1.93·11-s − 0.699·12-s − 0.0265·13-s + 1.00·14-s − 0.566·15-s − 1.24·16-s + 0.308·17-s − 1.48·18-s − 0.610·19-s + 0.177·20-s + 1.23·21-s + 2.35·22-s − 0.759·23-s − 0.961·24-s − 0.855·25-s + 0.0322·26-s − 0.341·27-s − 0.387·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2307303617\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2307303617\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 547 | \( 1 - T \) |
| good | 2 | \( 1 + 1.71T + 2T^{2} \) |
| 3 | \( 1 + 2.58T + 3T^{2} \) |
| 5 | \( 1 - 0.849T + 5T^{2} \) |
| 7 | \( 1 + 2.18T + 7T^{2} \) |
| 11 | \( 1 + 6.43T + 11T^{2} \) |
| 13 | \( 1 + 0.0958T + 13T^{2} \) |
| 17 | \( 1 - 1.27T + 17T^{2} \) |
| 19 | \( 1 + 2.66T + 19T^{2} \) |
| 23 | \( 1 + 3.64T + 23T^{2} \) |
| 29 | \( 1 - 0.787T + 29T^{2} \) |
| 31 | \( 1 + 1.39T + 31T^{2} \) |
| 37 | \( 1 - 8.37T + 37T^{2} \) |
| 41 | \( 1 - 5.17T + 41T^{2} \) |
| 43 | \( 1 - 9.89T + 43T^{2} \) |
| 47 | \( 1 - 4.41T + 47T^{2} \) |
| 53 | \( 1 - 10.2T + 53T^{2} \) |
| 59 | \( 1 - 6.44T + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 + 5.75T + 67T^{2} \) |
| 71 | \( 1 - 12.9T + 71T^{2} \) |
| 73 | \( 1 + 2.65T + 73T^{2} \) |
| 79 | \( 1 + 2.33T + 79T^{2} \) |
| 83 | \( 1 - 11.8T + 83T^{2} \) |
| 89 | \( 1 + 5.04T + 89T^{2} \) |
| 97 | \( 1 + 1.62T + 97T^{2} \) |
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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
−10.53611945165277114090543575559, −10.11699919337080856679372376248, −9.319635369736984656814770889738, −8.054423212377638345938741521943, −7.34456879242343680841834907298, −6.15494092818144482600350486564, −5.53639149053326177549066038215, −4.38329401549724452285926822110, −2.38566139156229888911961491081, −0.52609053459458882100432932239,
0.52609053459458882100432932239, 2.38566139156229888911961491081, 4.38329401549724452285926822110, 5.53639149053326177549066038215, 6.15494092818144482600350486564, 7.34456879242343680841834907298, 8.054423212377638345938741521943, 9.319635369736984656814770889738, 10.11699919337080856679372376248, 10.53611945165277114090543575559
