| L(s) = 1 | − 2·4-s − 5-s + 2·7-s + 3·11-s − 4·13-s + 4·16-s + 2·20-s + 6·23-s + 25-s − 4·28-s − 3·29-s + 5·31-s − 2·35-s + 8·37-s + 6·41-s − 4·43-s − 6·44-s − 6·47-s − 3·49-s + 8·52-s + 6·53-s − 3·55-s + 9·59-s − 7·61-s − 8·64-s + 4·65-s + 2·67-s + ⋯ |
| L(s) = 1 | − 4-s − 0.447·5-s + 0.755·7-s + 0.904·11-s − 1.10·13-s + 16-s + 0.447·20-s + 1.25·23-s + 1/5·25-s − 0.755·28-s − 0.557·29-s + 0.898·31-s − 0.338·35-s + 1.31·37-s + 0.937·41-s − 0.609·43-s − 0.904·44-s − 0.875·47-s − 3/7·49-s + 1.10·52-s + 0.824·53-s − 0.404·55-s + 1.17·59-s − 0.896·61-s − 64-s + 0.496·65-s + 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.642200492\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.642200492\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−15.94675851975323, −15.11576696509544, −14.62957584917452, −14.57785346445890, −13.77407725836977, −13.14433473016803, −12.64857328504626, −12.06751775812180, −11.40205023105670, −11.11606106612384, −10.06486911488800, −9.751071167173843, −9.022878681935540, −8.603654008920636, −7.891410810450812, −7.437900378069967, −6.708118713043401, −5.896216199735545, −5.053701339316527, −4.688248448006162, −4.107078241318526, −3.345346023685785, −2.490061879914128, −1.381426472687834, −0.6114223112306049,
0.6114223112306049, 1.381426472687834, 2.490061879914128, 3.345346023685785, 4.107078241318526, 4.688248448006162, 5.053701339316527, 5.896216199735545, 6.708118713043401, 7.437900378069967, 7.891410810450812, 8.603654008920636, 9.022878681935540, 9.751071167173843, 10.06486911488800, 11.11606106612384, 11.40205023105670, 12.06751775812180, 12.64857328504626, 13.14433473016803, 13.77407725836977, 14.57785346445890, 14.62957584917452, 15.11576696509544, 15.94675851975323
