| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 4·7-s − 8-s + 9-s + 10-s + 4·11-s + 12-s − 6·13-s − 4·14-s − 15-s + 16-s + 6·17-s − 18-s + 19-s − 20-s + 4·21-s − 4·22-s − 24-s + 25-s + 6·26-s + 27-s + 4·28-s + 6·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s + 0.288·12-s − 1.66·13-s − 1.06·14-s − 0.258·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.229·19-s − 0.223·20-s + 0.872·21-s − 0.852·22-s − 0.204·24-s + 1/5·25-s + 1.17·26-s + 0.192·27-s + 0.755·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 301530 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 301530 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.469671851\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.469671851\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 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
−12.38371793375757, −12.15343951406729, −11.86301105073542, −11.24773756816372, −10.88714609969843, −10.30752249477143, −9.897095599615790, −9.347392441460189, −8.986989072531011, −8.581255391808434, −7.963368150247029, −7.605492669923706, −7.274193901820847, −7.041784854364967, −6.047015528503603, −5.620676925680291, −5.057497484016709, −4.408837515729700, −4.141942854442337, −3.436896827655356, −2.743810368137677, −2.350563261263194, −1.590013532445563, −1.199238016720065, −0.5743184478923580,
0.5743184478923580, 1.199238016720065, 1.590013532445563, 2.350563261263194, 2.743810368137677, 3.436896827655356, 4.141942854442337, 4.408837515729700, 5.057497484016709, 5.620676925680291, 6.047015528503603, 7.041784854364967, 7.274193901820847, 7.605492669923706, 7.963368150247029, 8.581255391808434, 8.986989072531011, 9.347392441460189, 9.897095599615790, 10.30752249477143, 10.88714609969843, 11.24773756816372, 11.86301105073542, 12.15343951406729, 12.38371793375757
