| L(s) = 1 | + 0.700·2-s − 3-s − 1.50·4-s − 1.15·5-s − 0.700·6-s + 7-s − 2.45·8-s + 9-s − 0.809·10-s + 3.66·11-s + 1.50·12-s + 2.34·13-s + 0.700·14-s + 1.15·15-s + 1.29·16-s + 4.80·17-s + 0.700·18-s + 7.06·19-s + 1.74·20-s − 21-s + 2.56·22-s − 23-s + 2.45·24-s − 3.66·25-s + 1.64·26-s − 27-s − 1.50·28-s + ⋯ |
| L(s) = 1 | + 0.494·2-s − 0.577·3-s − 0.754·4-s − 0.517·5-s − 0.285·6-s + 0.377·7-s − 0.868·8-s + 0.333·9-s − 0.256·10-s + 1.10·11-s + 0.435·12-s + 0.650·13-s + 0.187·14-s + 0.298·15-s + 0.324·16-s + 1.16·17-s + 0.164·18-s + 1.62·19-s + 0.390·20-s − 0.218·21-s + 0.547·22-s − 0.208·23-s + 0.501·24-s − 0.732·25-s + 0.322·26-s − 0.192·27-s − 0.285·28-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
| \(L(1)\) |
\(\approx\) |
\(1.265563849\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.265563849\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.700T + 2T^{2} \) |
| 5 | \( 1 + 1.15T + 5T^{2} \) |
| 11 | \( 1 - 3.66T + 11T^{2} \) |
| 13 | \( 1 - 2.34T + 13T^{2} \) |
| 17 | \( 1 - 4.80T + 17T^{2} \) |
| 19 | \( 1 - 7.06T + 19T^{2} \) |
| 29 | \( 1 + 6.52T + 29T^{2} \) |
| 31 | \( 1 - 2.80T + 31T^{2} \) |
| 37 | \( 1 - 5.40T + 37T^{2} \) |
| 41 | \( 1 - 0.647T + 41T^{2} \) |
| 43 | \( 1 + 4.76T + 43T^{2} \) |
| 47 | \( 1 - 4.85T + 47T^{2} \) |
| 53 | \( 1 - 7.63T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 - 9.91T + 61T^{2} \) |
| 67 | \( 1 - 6.07T + 67T^{2} \) |
| 71 | \( 1 + 3.65T + 71T^{2} \) |
| 73 | \( 1 - 11.8T + 73T^{2} \) |
| 79 | \( 1 + 9.12T + 79T^{2} \) |
| 83 | \( 1 + 15.8T + 83T^{2} \) |
| 89 | \( 1 - 4.66T + 89T^{2} \) |
| 97 | \( 1 + 4.90T + 97T^{2} \) |
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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
−11.41542281002759819845602597215, −9.990255747821353218482091201633, −9.320812417925288448947507920263, −8.230549114441965978979785307526, −7.33753146973104786101675728606, −6.00430580177124077695535307342, −5.33080459027650471886209431621, −4.15548279852060529630866474558, −3.46742188186919482396754776148, −1.08236163856229215760881166669,
1.08236163856229215760881166669, 3.46742188186919482396754776148, 4.15548279852060529630866474558, 5.33080459027650471886209431621, 6.00430580177124077695535307342, 7.33753146973104786101675728606, 8.230549114441965978979785307526, 9.320812417925288448947507920263, 9.990255747821353218482091201633, 11.41542281002759819845602597215
