| L(s) = 1 | − 5-s − 4·7-s + 11-s + 5·13-s + 3·17-s + 3·23-s + 25-s + 8·29-s + 31-s + 4·35-s + 8·37-s + 41-s + 43-s + 9·49-s − 3·53-s − 55-s + 8·61-s − 5·65-s + 9·67-s + 4·71-s + 4·73-s − 4·77-s − 8·79-s + 15·83-s − 3·85-s + 16·89-s − 20·91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.51·7-s + 0.301·11-s + 1.38·13-s + 0.727·17-s + 0.625·23-s + 1/5·25-s + 1.48·29-s + 0.179·31-s + 0.676·35-s + 1.31·37-s + 0.156·41-s + 0.152·43-s + 9/7·49-s − 0.412·53-s − 0.134·55-s + 1.02·61-s − 0.620·65-s + 1.09·67-s + 0.474·71-s + 0.468·73-s − 0.455·77-s − 0.900·79-s + 1.64·83-s − 0.325·85-s + 1.69·89-s − 2.09·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.141713133\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.141713133\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 15 T + p T^{2} \) |
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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
−15.12893939837423, −14.64554341299169, −13.93860623591533, −13.44399899604114, −13.02568351663437, −12.41954526399456, −12.06073296049391, −11.31951656188640, −10.84326206097123, −10.27659852116985, −9.561459404235429, −9.350285578444364, −8.442016135992961, −8.195208996319893, −7.336298278589958, −6.664228868132583, −6.347699778721180, −5.778749155856392, −4.979659816777716, −4.147213885346132, −3.613529229864049, −3.119515250080586, −2.460771804966033, −1.168181197288951, −0.6567826164987082,
0.6567826164987082, 1.168181197288951, 2.460771804966033, 3.119515250080586, 3.613529229864049, 4.147213885346132, 4.979659816777716, 5.778749155856392, 6.347699778721180, 6.664228868132583, 7.336298278589958, 8.195208996319893, 8.442016135992961, 9.350285578444364, 9.561459404235429, 10.27659852116985, 10.84326206097123, 11.31951656188640, 12.06073296049391, 12.41954526399456, 13.02568351663437, 13.44399899604114, 13.93860623591533, 14.64554341299169, 15.12893939837423
