| L(s) = 1 | + 0.814·2-s + 1.40·3-s − 1.33·4-s + 1.41·5-s + 1.14·6-s + 3.63·7-s − 2.71·8-s − 1.03·9-s + 1.15·10-s − 1.41·11-s − 1.87·12-s + 4.78·13-s + 2.95·14-s + 1.98·15-s + 0.460·16-s + 4.74·17-s − 0.843·18-s − 0.219·19-s − 1.89·20-s + 5.08·21-s − 1.15·22-s + 8.30·23-s − 3.80·24-s − 2.99·25-s + 3.89·26-s − 5.65·27-s − 4.85·28-s + ⋯ |
| L(s) = 1 | + 0.575·2-s + 0.809·3-s − 0.668·4-s + 0.633·5-s + 0.465·6-s + 1.37·7-s − 0.960·8-s − 0.345·9-s + 0.364·10-s − 0.426·11-s − 0.540·12-s + 1.32·13-s + 0.790·14-s + 0.512·15-s + 0.115·16-s + 1.15·17-s − 0.198·18-s − 0.0504·19-s − 0.423·20-s + 1.11·21-s − 0.245·22-s + 1.73·23-s − 0.777·24-s − 0.598·25-s + 0.763·26-s − 1.08·27-s − 0.917·28-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
| \(L(1)\) |
\(\approx\) |
\(2.509331962\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.509331962\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.814T + 2T^{2} \) |
| 3 | \( 1 - 1.40T + 3T^{2} \) |
| 5 | \( 1 - 1.41T + 5T^{2} \) |
| 7 | \( 1 - 3.63T + 7T^{2} \) |
| 11 | \( 1 + 1.41T + 11T^{2} \) |
| 13 | \( 1 - 4.78T + 13T^{2} \) |
| 17 | \( 1 - 4.74T + 17T^{2} \) |
| 19 | \( 1 + 0.219T + 19T^{2} \) |
| 23 | \( 1 - 8.30T + 23T^{2} \) |
| 29 | \( 1 + 8.51T + 29T^{2} \) |
| 31 | \( 1 + 4.94T + 31T^{2} \) |
| 37 | \( 1 + 8.81T + 37T^{2} \) |
| 41 | \( 1 - 7.32T + 41T^{2} \) |
| 43 | \( 1 - 1.68T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 - 5.38T + 53T^{2} \) |
| 59 | \( 1 - 9.25T + 59T^{2} \) |
| 61 | \( 1 - 8.39T + 61T^{2} \) |
| 67 | \( 1 + 6.01T + 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 + 9.04T + 73T^{2} \) |
| 79 | \( 1 + 8.82T + 79T^{2} \) |
| 83 | \( 1 + 0.697T + 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 + 13.9T + 97T^{2} \) |
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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
−10.93634059411833847923959427500, −9.729799824388761606633071579758, −8.835140814202594975964733594552, −8.384440920849576959474709602871, −7.39980525741618121959544629302, −5.63298851995616045873602865105, −5.40618109907081804054038395905, −4.00832219078500144892569689504, −3.07648014145653002981647761030, −1.59876633439743047490432555229,
1.59876633439743047490432555229, 3.07648014145653002981647761030, 4.00832219078500144892569689504, 5.40618109907081804054038395905, 5.63298851995616045873602865105, 7.39980525741618121959544629302, 8.384440920849576959474709602871, 8.835140814202594975964733594552, 9.729799824388761606633071579758, 10.93634059411833847923959427500
