| L(s) = 1 | − 5·7-s − 5·11-s + 13-s + 3·17-s − 4·19-s − 5·23-s + 4·29-s − 7·37-s − 11·41-s − 12·43-s − 6·47-s + 18·49-s − 53-s − 12·59-s − 7·61-s + 4·67-s + 7·71-s − 14·73-s + 25·77-s − 5·79-s + 2·83-s + 3·89-s − 5·91-s − 97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 1.88·7-s − 1.50·11-s + 0.277·13-s + 0.727·17-s − 0.917·19-s − 1.04·23-s + 0.742·29-s − 1.15·37-s − 1.71·41-s − 1.82·43-s − 0.875·47-s + 18/7·49-s − 0.137·53-s − 1.56·59-s − 0.896·61-s + 0.488·67-s + 0.830·71-s − 1.63·73-s + 2.84·77-s − 0.562·79-s + 0.219·83-s + 0.317·89-s − 0.524·91-s − 0.101·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−15.97723975598883, −15.51669423746149, −15.17339873561220, −14.30597819394445, −13.60178503132281, −13.31095726740083, −12.85721073640612, −12.16070549893152, −11.97847739898669, −10.87400132778325, −10.29586187781508, −10.12215903189900, −9.568189484041309, −8.729499878883096, −8.256533533012312, −7.669347750557138, −6.799087832492371, −6.493579426510067, −5.826283447091323, −5.223221105654956, −4.485779145043619, −3.461393944515141, −3.248355936637977, −2.476925948883766, −1.570907241953605, 0, 0,
1.570907241953605, 2.476925948883766, 3.248355936637977, 3.461393944515141, 4.485779145043619, 5.223221105654956, 5.826283447091323, 6.493579426510067, 6.799087832492371, 7.669347750557138, 8.256533533012312, 8.729499878883096, 9.568189484041309, 10.12215903189900, 10.29586187781508, 10.87400132778325, 11.97847739898669, 12.16070549893152, 12.85721073640612, 13.31095726740083, 13.60178503132281, 14.30597819394445, 15.17339873561220, 15.51669423746149, 15.97723975598883
