| L(s) = 1 | − 2·4-s + 3·5-s − 7-s + 11-s + 4·16-s − 2·17-s − 6·20-s + 4·23-s + 4·25-s + 2·28-s − 5·31-s − 3·35-s + 10·37-s − 5·41-s + 4·43-s − 2·44-s − 3·47-s + 49-s − 5·53-s + 3·55-s + 10·59-s − 5·61-s − 8·64-s − 10·67-s + 4·68-s − 15·71-s + 14·73-s + ⋯ |
| L(s) = 1 | − 4-s + 1.34·5-s − 0.377·7-s + 0.301·11-s + 16-s − 0.485·17-s − 1.34·20-s + 0.834·23-s + 4/5·25-s + 0.377·28-s − 0.898·31-s − 0.507·35-s + 1.64·37-s − 0.780·41-s + 0.609·43-s − 0.301·44-s − 0.437·47-s + 1/7·49-s − 0.686·53-s + 0.404·55-s + 1.30·59-s − 0.640·61-s − 64-s − 1.22·67-s + 0.485·68-s − 1.78·71-s + 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 250173 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250173 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.084894752\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.084894752\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−13.10613543898084, −12.52071991582358, −12.10959529360645, −11.35543099480657, −10.97945271349636, −10.32475459809702, −10.05244193683571, −9.446575365652436, −9.235252677522015, −8.880706092521521, −8.312710782962611, −7.672101691281071, −7.207683023922313, −6.418496780677454, −6.265370426471996, −5.594702934908881, −5.246021040381126, −4.671021001492597, −4.182701295026627, −3.567306001355160, −2.964466341587790, −2.423519651468007, −1.695529726155318, −1.176615074191534, −0.4200599447314332,
0.4200599447314332, 1.176615074191534, 1.695529726155318, 2.423519651468007, 2.964466341587790, 3.567306001355160, 4.182701295026627, 4.671021001492597, 5.246021040381126, 5.594702934908881, 6.265370426471996, 6.418496780677454, 7.207683023922313, 7.672101691281071, 8.312710782962611, 8.880706092521521, 9.235252677522015, 9.446575365652436, 10.05244193683571, 10.32475459809702, 10.97945271349636, 11.35543099480657, 12.10959529360645, 12.52071991582358, 13.10613543898084
