| L(s) = 1 | + 3.58·5-s + 7-s − 2.81·11-s + 13-s + 4.11·17-s + 0.888·19-s + 5.87·23-s + 7.87·25-s + 1.69·29-s + 6.98·31-s + 3.58·35-s + 4.76·37-s + 5.41·41-s − 5.21·43-s − 2.66·47-s + 49-s + 0.123·53-s − 10.0·55-s − 8.87·59-s + 3.87·61-s + 3.58·65-s − 12.3·67-s − 2.87·71-s − 10.6·73-s − 2.81·77-s + 7.08·79-s − 4.11·83-s + ⋯ |
| L(s) = 1 | + 1.60·5-s + 0.377·7-s − 0.847·11-s + 0.277·13-s + 0.997·17-s + 0.203·19-s + 1.22·23-s + 1.57·25-s + 0.315·29-s + 1.25·31-s + 0.606·35-s + 0.783·37-s + 0.845·41-s − 0.794·43-s − 0.388·47-s + 0.142·49-s + 0.0169·53-s − 1.36·55-s − 1.15·59-s + 0.496·61-s + 0.445·65-s − 1.50·67-s − 0.341·71-s − 1.24·73-s − 0.320·77-s + 0.797·79-s − 0.451·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.420327907\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.420327907\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 - 3.58T + 5T^{2} \) |
| 11 | \( 1 + 2.81T + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 - 4.11T + 17T^{2} \) |
| 19 | \( 1 - 0.888T + 19T^{2} \) |
| 23 | \( 1 - 5.87T + 23T^{2} \) |
| 29 | \( 1 - 1.69T + 29T^{2} \) |
| 31 | \( 1 - 6.98T + 31T^{2} \) |
| 37 | \( 1 - 4.76T + 37T^{2} \) |
| 41 | \( 1 - 5.41T + 41T^{2} \) |
| 43 | \( 1 + 5.21T + 43T^{2} \) |
| 47 | \( 1 + 2.66T + 47T^{2} \) |
| 53 | \( 1 - 0.123T + 53T^{2} \) |
| 59 | \( 1 + 8.87T + 59T^{2} \) |
| 61 | \( 1 - 3.87T + 61T^{2} \) |
| 67 | \( 1 + 12.3T + 67T^{2} \) |
| 71 | \( 1 + 2.87T + 71T^{2} \) |
| 73 | \( 1 + 10.6T + 73T^{2} \) |
| 79 | \( 1 - 7.08T + 79T^{2} \) |
| 83 | \( 1 + 4.11T + 83T^{2} \) |
| 89 | \( 1 + 9.60T + 89T^{2} \) |
| 97 | \( 1 - 7.32T + 97T^{2} \) |
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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
−7.75546180313068116135228470335, −7.00578373168963814026370086657, −6.17795154977534849589422768204, −5.72626227360949152012655161868, −5.06238144075881043186050596758, −4.48474285735323855261820681894, −3.11088610112192956194849862180, −2.69449569216128473217024340306, −1.68632252229957730632698970736, −0.958886695230067146015558401349,
0.958886695230067146015558401349, 1.68632252229957730632698970736, 2.69449569216128473217024340306, 3.11088610112192956194849862180, 4.48474285735323855261820681894, 5.06238144075881043186050596758, 5.72626227360949152012655161868, 6.17795154977534849589422768204, 7.00578373168963814026370086657, 7.75546180313068116135228470335
