| L(s) = 1 | − 3-s − 7-s + 9-s + 5·11-s + 13-s − 3·17-s + 21-s − 4·23-s − 27-s − 9·29-s + 7·31-s − 5·33-s − 8·37-s − 39-s − 2·41-s + 7·47-s − 6·49-s + 3·51-s − 3·53-s − 9·59-s + 15·61-s − 63-s + 7·67-s + 4·69-s + 4·73-s − 5·77-s + 8·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s + 1.50·11-s + 0.277·13-s − 0.727·17-s + 0.218·21-s − 0.834·23-s − 0.192·27-s − 1.67·29-s + 1.25·31-s − 0.870·33-s − 1.31·37-s − 0.160·39-s − 0.312·41-s + 1.02·47-s − 6/7·49-s + 0.420·51-s − 0.412·53-s − 1.17·59-s + 1.92·61-s − 0.125·63-s + 0.855·67-s + 0.481·69-s + 0.468·73-s − 0.569·77-s + 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 15 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−14.29330263344288, −14.15572487509239, −13.48875705400350, −12.95168530214795, −12.46029129659568, −11.94392098495110, −11.45239782093973, −11.17412707844559, −10.42446553675046, −9.961231295687914, −9.388897399346677, −8.984060275703562, −8.393685061262288, −7.766002084537785, −6.966871961972292, −6.707956058980424, −6.136757588382768, −5.683655455310390, −4.933374902624748, −4.301878930405018, −3.767228347110515, −3.324146247976971, −2.224553001757119, −1.700062759363043, −0.8785380854951716, 0,
0.8785380854951716, 1.700062759363043, 2.224553001757119, 3.324146247976971, 3.767228347110515, 4.301878930405018, 4.933374902624748, 5.683655455310390, 6.136757588382768, 6.707956058980424, 6.966871961972292, 7.766002084537785, 8.393685061262288, 8.984060275703562, 9.388897399346677, 9.961231295687914, 10.42446553675046, 11.17412707844559, 11.45239782093973, 11.94392098495110, 12.46029129659568, 12.95168530214795, 13.48875705400350, 14.15572487509239, 14.29330263344288
