| L(s) = 1 | + 0.731·2-s − 2.97·3-s − 1.46·4-s + 2.76·5-s − 2.17·6-s − 1.34·7-s − 2.53·8-s + 5.85·9-s + 2.02·10-s − 11-s + 4.35·12-s − 4.81·13-s − 0.985·14-s − 8.22·15-s + 1.07·16-s + 6.15·17-s + 4.28·18-s − 0.496·19-s − 4.04·20-s + 4.00·21-s − 0.731·22-s + 7.44·23-s + 7.54·24-s + 2.64·25-s − 3.52·26-s − 8.49·27-s + 1.97·28-s + ⋯ |
| L(s) = 1 | + 0.517·2-s − 1.71·3-s − 0.732·4-s + 1.23·5-s − 0.889·6-s − 0.508·7-s − 0.896·8-s + 1.95·9-s + 0.639·10-s − 0.301·11-s + 1.25·12-s − 1.33·13-s − 0.263·14-s − 2.12·15-s + 0.268·16-s + 1.49·17-s + 1.00·18-s − 0.113·19-s − 0.905·20-s + 0.874·21-s − 0.156·22-s + 1.55·23-s + 1.54·24-s + 0.528·25-s − 0.690·26-s − 1.63·27-s + 0.372·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9488515009\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9488515009\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 - 0.731T + 2T^{2} \) |
| 3 | \( 1 + 2.97T + 3T^{2} \) |
| 5 | \( 1 - 2.76T + 5T^{2} \) |
| 7 | \( 1 + 1.34T + 7T^{2} \) |
| 13 | \( 1 + 4.81T + 13T^{2} \) |
| 17 | \( 1 - 6.15T + 17T^{2} \) |
| 19 | \( 1 + 0.496T + 19T^{2} \) |
| 23 | \( 1 - 7.44T + 23T^{2} \) |
| 29 | \( 1 + 0.647T + 29T^{2} \) |
| 31 | \( 1 - 4.67T + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 - 9.09T + 41T^{2} \) |
| 43 | \( 1 + 9.64T + 43T^{2} \) |
| 47 | \( 1 - 8.36T + 47T^{2} \) |
| 53 | \( 1 - 9.40T + 53T^{2} \) |
| 59 | \( 1 - 0.756T + 59T^{2} \) |
| 67 | \( 1 + 9.70T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 - 4.46T + 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 - 16.8T + 89T^{2} \) |
| 97 | \( 1 + 2.10T + 97T^{2} \) |
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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
−10.29164609489742233511013750594, −9.917883301619932491842692686095, −9.216101485319938753050461032194, −7.58971086125204050270189287180, −6.51260354729613177890815036809, −5.72896835081996095282376587818, −5.26556745138493777679008247057, −4.46422027428884276143689270447, −2.83180658525507084349702004104, −0.851104959327967857128684774384,
0.851104959327967857128684774384, 2.83180658525507084349702004104, 4.46422027428884276143689270447, 5.26556745138493777679008247057, 5.72896835081996095282376587818, 6.51260354729613177890815036809, 7.58971086125204050270189287180, 9.216101485319938753050461032194, 9.917883301619932491842692686095, 10.29164609489742233511013750594
