| L(s) = 1 | − 5-s + 2·7-s − 11-s − 13-s − 2·17-s − 4·19-s + 8·23-s + 25-s − 4·29-s + 10·31-s − 2·35-s − 4·37-s − 6·41-s + 4·43-s + 8·47-s − 3·49-s − 12·53-s + 55-s − 2·61-s + 65-s + 8·67-s + 8·71-s + 2·73-s − 2·77-s + 8·79-s + 4·83-s + 2·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.755·7-s − 0.301·11-s − 0.277·13-s − 0.485·17-s − 0.917·19-s + 1.66·23-s + 1/5·25-s − 0.742·29-s + 1.79·31-s − 0.338·35-s − 0.657·37-s − 0.937·41-s + 0.609·43-s + 1.16·47-s − 3/7·49-s − 1.64·53-s + 0.134·55-s − 0.256·61-s + 0.124·65-s + 0.977·67-s + 0.949·71-s + 0.234·73-s − 0.227·77-s + 0.900·79-s + 0.439·83-s + 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.889148728\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.889148728\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−13.83810786263642, −13.18839357435886, −12.67534917681959, −12.36499752322447, −11.67640023850488, −11.24122160046197, −10.86391443838903, −10.44901862867238, −9.785600479467727, −9.170058079099207, −8.736116934335917, −8.198513254212507, −7.814790668409170, −7.245108803810284, −6.595405080572814, −6.341824488618342, −5.268869549917281, −5.080319723551836, −4.469671551046447, −3.964818697108323, −3.195825584373380, −2.622730538131639, −1.988180218766960, −1.238909286628911, −0.4497122916981749,
0.4497122916981749, 1.238909286628911, 1.988180218766960, 2.622730538131639, 3.195825584373380, 3.964818697108323, 4.469671551046447, 5.080319723551836, 5.268869549917281, 6.341824488618342, 6.595405080572814, 7.245108803810284, 7.814790668409170, 8.198513254212507, 8.736116934335917, 9.170058079099207, 9.785600479467727, 10.44901862867238, 10.86391443838903, 11.24122160046197, 11.67640023850488, 12.36499752322447, 12.67534917681959, 13.18839357435886, 13.83810786263642
