| L(s) = 1 | − 3-s + 7-s − 2·9-s + 11-s + 13-s + 17-s − 2·19-s − 21-s − 23-s + 5·27-s + 7·29-s − 4·31-s − 33-s + 8·37-s − 39-s − 6·41-s + 8·43-s + 7·47-s + 49-s − 51-s + 4·53-s + 2·57-s + 4·59-s − 10·61-s − 2·63-s + 14·67-s + 69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s − 2/3·9-s + 0.301·11-s + 0.277·13-s + 0.242·17-s − 0.458·19-s − 0.218·21-s − 0.208·23-s + 0.962·27-s + 1.29·29-s − 0.718·31-s − 0.174·33-s + 1.31·37-s − 0.160·39-s − 0.937·41-s + 1.21·43-s + 1.02·47-s + 1/7·49-s − 0.140·51-s + 0.549·53-s + 0.264·57-s + 0.520·59-s − 1.28·61-s − 0.251·63-s + 1.71·67-s + 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.984005431\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.984005431\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−14.23711736124725, −13.77254609466009, −13.29281678298830, −12.54665612132162, −12.12208034646403, −11.80947691487956, −11.13541287007282, −10.72686329264173, −10.43767172746399, −9.514334110865576, −9.210161118375902, −8.469339380081694, −8.124665723352248, −7.556074743879272, −6.691315513413977, −6.462750522186132, −5.696867744706072, −5.391700986341262, −4.650119490303055, −4.124213523339483, −3.455556815802171, −2.668048971507700, −2.127525913765752, −1.134436370341860, −0.5593200447105998,
0.5593200447105998, 1.134436370341860, 2.127525913765752, 2.668048971507700, 3.455556815802171, 4.124213523339483, 4.650119490303055, 5.391700986341262, 5.696867744706072, 6.462750522186132, 6.691315513413977, 7.556074743879272, 8.124665723352248, 8.469339380081694, 9.210161118375902, 9.514334110865576, 10.43767172746399, 10.72686329264173, 11.13541287007282, 11.80947691487956, 12.12208034646403, 12.54665612132162, 13.29281678298830, 13.77254609466009, 14.23711736124725
