| L(s) = 1 | + 5-s + 2·11-s + 13-s − 4·17-s − 4·19-s − 4·23-s + 25-s + 2·29-s + 2·31-s + 6·37-s − 2·43-s − 8·47-s + 12·53-s + 2·55-s − 4·59-s + 2·61-s + 65-s − 16·67-s + 12·71-s − 6·73-s − 4·83-s − 4·85-s − 4·95-s − 10·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.603·11-s + 0.277·13-s − 0.970·17-s − 0.917·19-s − 0.834·23-s + 1/5·25-s + 0.371·29-s + 0.359·31-s + 0.986·37-s − 0.304·43-s − 1.16·47-s + 1.64·53-s + 0.269·55-s − 0.520·59-s + 0.256·61-s + 0.124·65-s − 1.95·67-s + 1.42·71-s − 0.702·73-s − 0.439·83-s − 0.433·85-s − 0.410·95-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.884747732\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.884747732\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−13.04592043472865, −12.53310508293875, −11.80354311121465, −11.73716475909589, −10.99075880018448, −10.65727379258804, −10.11019524696887, −9.696085919365570, −9.178686573044467, −8.661361992010538, −8.377143094330508, −7.780699911341620, −7.139019933753257, −6.638848908309720, −6.190953602786407, −5.954165743790058, −5.132526664757264, −4.654476712444640, −4.074851724010976, −3.771384706766247, −2.839935433386775, −2.454154643666637, −1.772801408296169, −1.270016984672218, −0.3758071111245694,
0.3758071111245694, 1.270016984672218, 1.772801408296169, 2.454154643666637, 2.839935433386775, 3.771384706766247, 4.074851724010976, 4.654476712444640, 5.132526664757264, 5.954165743790058, 6.190953602786407, 6.638848908309720, 7.139019933753257, 7.780699911341620, 8.377143094330508, 8.661361992010538, 9.178686573044467, 9.696085919365570, 10.11019524696887, 10.65727379258804, 10.99075880018448, 11.73716475909589, 11.80354311121465, 12.53310508293875, 13.04592043472865
