| L(s) = 1 | − 0.509·2-s − 3-s − 1.74·4-s + 4.41·5-s + 0.509·6-s + 7-s + 1.90·8-s + 9-s − 2.24·10-s − 1.67·11-s + 1.74·12-s − 4.66·13-s − 0.509·14-s − 4.41·15-s + 2.50·16-s + 6.24·17-s − 0.509·18-s − 0.694·19-s − 7.68·20-s − 21-s + 0.853·22-s − 23-s − 1.90·24-s + 14.4·25-s + 2.37·26-s − 27-s − 1.74·28-s + ⋯ |
| L(s) = 1 | − 0.360·2-s − 0.577·3-s − 0.870·4-s + 1.97·5-s + 0.208·6-s + 0.377·7-s + 0.673·8-s + 0.333·9-s − 0.711·10-s − 0.505·11-s + 0.502·12-s − 1.29·13-s − 0.136·14-s − 1.14·15-s + 0.627·16-s + 1.51·17-s − 0.120·18-s − 0.159·19-s − 1.71·20-s − 0.218·21-s + 0.181·22-s − 0.208·23-s − 0.389·24-s + 2.89·25-s + 0.466·26-s − 0.192·27-s − 0.328·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.162459983\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.162459983\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 0.509T + 2T^{2} \) |
| 5 | \( 1 - 4.41T + 5T^{2} \) |
| 11 | \( 1 + 1.67T + 11T^{2} \) |
| 13 | \( 1 + 4.66T + 13T^{2} \) |
| 17 | \( 1 - 6.24T + 17T^{2} \) |
| 19 | \( 1 + 0.694T + 19T^{2} \) |
| 29 | \( 1 - 5.60T + 29T^{2} \) |
| 31 | \( 1 - 4.24T + 31T^{2} \) |
| 37 | \( 1 - 9.26T + 37T^{2} \) |
| 41 | \( 1 + 5.15T + 41T^{2} \) |
| 43 | \( 1 - 4.20T + 43T^{2} \) |
| 47 | \( 1 + 1.92T + 47T^{2} \) |
| 53 | \( 1 + 1.84T + 53T^{2} \) |
| 59 | \( 1 - 9.39T + 59T^{2} \) |
| 61 | \( 1 - 7.72T + 61T^{2} \) |
| 67 | \( 1 + 8.22T + 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 + 11.9T + 73T^{2} \) |
| 79 | \( 1 - 0.581T + 79T^{2} \) |
| 83 | \( 1 - 6.95T + 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 - 10.0T + 97T^{2} \) |
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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
−10.45017466706392742399700860894, −10.04598277973823165877011393715, −9.543550628222837975990352195735, −8.428568299598721889872542319605, −7.37391490704649656117541285021, −6.08533705941490201296588103865, −5.31379707727142957389546270355, −4.67908851655242500978606384479, −2.62631240960185589906561375320, −1.19695069786031820778788067855,
1.19695069786031820778788067855, 2.62631240960185589906561375320, 4.67908851655242500978606384479, 5.31379707727142957389546270355, 6.08533705941490201296588103865, 7.37391490704649656117541285021, 8.428568299598721889872542319605, 9.543550628222837975990352195735, 10.04598277973823165877011393715, 10.45017466706392742399700860894
