| L(s) = 1 | + 3-s − 2·5-s − 7-s + 9-s + 4·11-s + 6·13-s − 2·15-s − 2·17-s + 4·19-s − 21-s + 4·23-s − 25-s + 27-s + 6·29-s − 8·31-s + 4·33-s + 2·35-s + 10·37-s + 6·39-s − 2·41-s + 12·43-s − 2·45-s + 8·47-s + 49-s − 2·51-s + 2·53-s − 8·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s + 1.66·13-s − 0.516·15-s − 0.485·17-s + 0.917·19-s − 0.218·21-s + 0.834·23-s − 1/5·25-s + 0.192·27-s + 1.11·29-s − 1.43·31-s + 0.696·33-s + 0.338·35-s + 1.64·37-s + 0.960·39-s − 0.312·41-s + 1.82·43-s − 0.298·45-s + 1.16·47-s + 1/7·49-s − 0.280·51-s + 0.274·53-s − 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.671066870\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.671066870\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 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
−14.39659939999387, −13.79946019306142, −13.53520790321811, −12.84873844072129, −12.41934416438463, −11.75185385198359, −11.38680230363391, −10.87311153505713, −10.40725110145866, −9.450121379338996, −9.121488748841623, −8.871681410087309, −8.077869774986694, −7.676498889818547, −7.074902277283568, −6.500335871740761, −6.027948609331566, −5.324706573427574, −4.332777799531621, −4.038049421193060, −3.522415317810369, −2.989630135380228, −2.141876927810831, −1.152076411205215, −0.7663314267691434,
0.7663314267691434, 1.152076411205215, 2.141876927810831, 2.989630135380228, 3.522415317810369, 4.038049421193060, 4.332777799531621, 5.324706573427574, 6.027948609331566, 6.500335871740761, 7.074902277283568, 7.676498889818547, 8.077869774986694, 8.871681410087309, 9.121488748841623, 9.450121379338996, 10.40725110145866, 10.87311153505713, 11.38680230363391, 11.75185385198359, 12.41934416438463, 12.84873844072129, 13.53520790321811, 13.79946019306142, 14.39659939999387
