| L(s) = 1 | − 5-s − 7-s + 4·11-s + 6·13-s − 2·17-s + 25-s + 6·29-s + 8·31-s + 35-s + 10·37-s − 2·41-s − 4·43-s − 8·47-s + 49-s − 2·53-s − 4·55-s − 8·59-s + 14·61-s − 6·65-s + 12·67-s + 16·71-s + 2·73-s − 4·77-s − 8·79-s + 8·83-s + 2·85-s − 10·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s + 1.20·11-s + 1.66·13-s − 0.485·17-s + 1/5·25-s + 1.11·29-s + 1.43·31-s + 0.169·35-s + 1.64·37-s − 0.312·41-s − 0.609·43-s − 1.16·47-s + 1/7·49-s − 0.274·53-s − 0.539·55-s − 1.04·59-s + 1.79·61-s − 0.744·65-s + 1.46·67-s + 1.89·71-s + 0.234·73-s − 0.455·77-s − 0.900·79-s + 0.878·83-s + 0.216·85-s − 1.05·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.559505658\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.559505658\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−15.63849773876631, −15.31626516797089, −14.43240008067825, −14.09870222465945, −13.43180387543223, −12.98689996589128, −12.36439548108731, −11.60635213091742, −11.40534296912298, −10.81553470467139, −9.981567531318383, −9.582689701167671, −8.797071991058377, −8.373819634415930, −7.930561923914175, −6.877945442645915, −6.389003823542150, −6.232040771882118, −5.124690347904054, −4.404317255567873, −3.830536110816996, −3.298957216991220, −2.433790063196622, −1.336676027019226, −0.7465416445234451,
0.7465416445234451, 1.336676027019226, 2.433790063196622, 3.298957216991220, 3.830536110816996, 4.404317255567873, 5.124690347904054, 6.232040771882118, 6.389003823542150, 6.877945442645915, 7.930561923914175, 8.373819634415930, 8.797071991058377, 9.582689701167671, 9.981567531318383, 10.81553470467139, 11.40534296912298, 11.60635213091742, 12.36439548108731, 12.98689996589128, 13.43180387543223, 14.09870222465945, 14.43240008067825, 15.31626516797089, 15.63849773876631
