| L(s) = 1 | + 5-s − 2·7-s − 4·13-s + 7·17-s − 4·19-s − 7·23-s + 25-s − 8·29-s − 7·31-s − 2·35-s + 10·37-s + 2·41-s − 9·47-s − 3·49-s − 9·53-s − 12·59-s + 61-s − 4·65-s − 2·67-s − 12·71-s − 8·73-s − 15·79-s + 7·85-s + 18·89-s + 8·91-s − 4·95-s + 14·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s − 1.10·13-s + 1.69·17-s − 0.917·19-s − 1.45·23-s + 1/5·25-s − 1.48·29-s − 1.25·31-s − 0.338·35-s + 1.64·37-s + 0.312·41-s − 1.31·47-s − 3/7·49-s − 1.23·53-s − 1.56·59-s + 0.128·61-s − 0.496·65-s − 0.244·67-s − 1.42·71-s − 0.936·73-s − 1.68·79-s + 0.759·85-s + 1.90·89-s + 0.838·91-s − 0.410·95-s + 1.42·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8433791620\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8433791620\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 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
−14.69889993365115, −14.41424846522188, −13.58399263993639, −12.97133144162513, −12.74055704542774, −12.20507257215128, −11.63242146873471, −11.02408687091157, −10.34020715980950, −9.873814289938798, −9.564779836643767, −9.087278882222884, −8.230124675390689, −7.531869551251958, −7.475373772214893, −6.444954948049937, −5.957119326609455, −5.655186210572350, −4.781379086025387, −4.232840959831821, −3.385833000071095, −2.994463163518389, −2.054557840676826, −1.588579430592121, −0.3125728741710219,
0.3125728741710219, 1.588579430592121, 2.054557840676826, 2.994463163518389, 3.385833000071095, 4.232840959831821, 4.781379086025387, 5.655186210572350, 5.957119326609455, 6.444954948049937, 7.475373772214893, 7.531869551251958, 8.230124675390689, 9.087278882222884, 9.564779836643767, 9.873814289938798, 10.34020715980950, 11.02408687091157, 11.63242146873471, 12.20507257215128, 12.74055704542774, 12.97133144162513, 13.58399263993639, 14.41424846522188, 14.69889993365115
