| L(s) = 1 | − 2·5-s + 7-s + 4·11-s − 6·13-s − 2·17-s − 4·19-s + 8·23-s − 25-s − 2·29-s − 2·35-s + 10·37-s + 6·41-s − 4·43-s + 49-s + 6·53-s − 8·55-s − 4·59-s − 6·61-s + 12·65-s + 4·67-s + 8·71-s + 10·73-s + 4·77-s + 4·83-s + 4·85-s + 6·89-s − 6·91-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.377·7-s + 1.20·11-s − 1.66·13-s − 0.485·17-s − 0.917·19-s + 1.66·23-s − 1/5·25-s − 0.371·29-s − 0.338·35-s + 1.64·37-s + 0.937·41-s − 0.609·43-s + 1/7·49-s + 0.824·53-s − 1.07·55-s − 0.520·59-s − 0.768·61-s + 1.48·65-s + 0.488·67-s + 0.949·71-s + 1.17·73-s + 0.455·77-s + 0.439·83-s + 0.433·85-s + 0.635·89-s − 0.628·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.418845483\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.418845483\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−18.03762756715260, −17.34674386537838, −16.86461218402957, −16.50017169365781, −15.37769240296599, −14.96621205207054, −14.68598176721100, −13.85948642424766, −12.96840492476808, −12.40878376259858, −11.80956405785084, −11.22075297586092, −10.72252892462507, −9.575429059410093, −9.258735862864324, −8.357312414327508, −7.687138909555049, −7.037619494193415, −6.437916833982970, −5.322401419345924, −4.490916403668538, −4.068024435551992, −2.950754869354144, −2.013763634717605, −0.6738920945868243,
0.6738920945868243, 2.013763634717605, 2.950754869354144, 4.068024435551992, 4.490916403668538, 5.322401419345924, 6.437916833982970, 7.037619494193415, 7.687138909555049, 8.357312414327508, 9.258735862864324, 9.575429059410093, 10.72252892462507, 11.22075297586092, 11.80956405785084, 12.40878376259858, 12.96840492476808, 13.85948642424766, 14.68598176721100, 14.96621205207054, 15.37769240296599, 16.50017169365781, 16.86461218402957, 17.34674386537838, 18.03762756715260
