| L(s) = 1 | + 5-s + 7-s + 4·11-s + 2·13-s + 2·17-s + 25-s + 6·29-s + 8·31-s + 35-s − 2·37-s + 6·41-s − 4·43-s + 49-s − 6·53-s + 4·55-s + 8·59-s + 2·61-s + 2·65-s − 4·67-s + 4·71-s − 10·73-s + 4·77-s + 12·79-s + 4·83-s + 2·85-s + 6·89-s + 2·91-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s + 1.20·11-s + 0.554·13-s + 0.485·17-s + 1/5·25-s + 1.11·29-s + 1.43·31-s + 0.169·35-s − 0.328·37-s + 0.937·41-s − 0.609·43-s + 1/7·49-s − 0.824·53-s + 0.539·55-s + 1.04·59-s + 0.256·61-s + 0.248·65-s − 0.488·67-s + 0.474·71-s − 1.17·73-s + 0.455·77-s + 1.35·79-s + 0.439·83-s + 0.216·85-s + 0.635·89-s + 0.209·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.489352557\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.489352557\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−15.77405858080465, −14.95374291940076, −14.54075896323626, −14.00473346181146, −13.62162515510243, −12.98075212718670, −12.19167464844979, −11.90766431251089, −11.28299148373855, −10.66034758382707, −10.05586647940757, −9.560706324776048, −8.883256725137914, −8.411982202377847, −7.817957858817457, −6.999606058482131, −6.391331822517310, −6.037607917086515, −5.157580371663007, −4.564769616217301, −3.871710061645515, −3.154965419340811, −2.331586576356375, −1.419770961867914, −0.8649855917463965,
0.8649855917463965, 1.419770961867914, 2.331586576356375, 3.154965419340811, 3.871710061645515, 4.564769616217301, 5.157580371663007, 6.037607917086515, 6.391331822517310, 6.999606058482131, 7.817957858817457, 8.411982202377847, 8.883256725137914, 9.560706324776048, 10.05586647940757, 10.66034758382707, 11.28299148373855, 11.90766431251089, 12.19167464844979, 12.98075212718670, 13.62162515510243, 14.00473346181146, 14.54075896323626, 14.95374291940076, 15.77405858080465
