| L(s) = 1 | − 1.19·5-s − 0.613·7-s − 1.53·11-s + 1.24·13-s + 4.35·17-s − 1.29·19-s − 4·23-s − 3.56·25-s − 0.965·29-s + 6.77·31-s + 0.735·35-s + 10.8·37-s − 8.68·41-s + 8.68·43-s − 9.65·47-s − 6.62·49-s + 11.4·53-s + 1.83·55-s − 9.04·59-s + 1.52·61-s − 1.49·65-s − 12.1·67-s − 3.56·71-s + 5.15·73-s + 0.938·77-s − 7.49·79-s + 7.18·83-s + ⋯ |
| L(s) = 1 | − 0.536·5-s − 0.231·7-s − 0.461·11-s + 0.346·13-s + 1.05·17-s − 0.297·19-s − 0.834·23-s − 0.712·25-s − 0.179·29-s + 1.21·31-s + 0.124·35-s + 1.77·37-s − 1.35·41-s + 1.32·43-s − 1.40·47-s − 0.946·49-s + 1.56·53-s + 0.247·55-s − 1.17·59-s + 0.195·61-s − 0.185·65-s − 1.47·67-s − 0.423·71-s + 0.603·73-s + 0.106·77-s − 0.843·79-s + 0.788·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.433604444\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.433604444\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 1.19T + 5T^{2} \) |
| 7 | \( 1 + 0.613T + 7T^{2} \) |
| 11 | \( 1 + 1.53T + 11T^{2} \) |
| 13 | \( 1 - 1.24T + 13T^{2} \) |
| 17 | \( 1 - 4.35T + 17T^{2} \) |
| 19 | \( 1 + 1.29T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 0.965T + 29T^{2} \) |
| 31 | \( 1 - 6.77T + 31T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 + 8.68T + 41T^{2} \) |
| 43 | \( 1 - 8.68T + 43T^{2} \) |
| 47 | \( 1 + 9.65T + 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 + 9.04T + 59T^{2} \) |
| 61 | \( 1 - 1.52T + 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 + 3.56T + 71T^{2} \) |
| 73 | \( 1 - 5.15T + 73T^{2} \) |
| 79 | \( 1 + 7.49T + 79T^{2} \) |
| 83 | \( 1 - 7.18T + 83T^{2} \) |
| 89 | \( 1 - 0.672T + 89T^{2} \) |
| 97 | \( 1 - 4.71T + 97T^{2} \) |
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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
−7.901636505504726285489337479754, −7.14771414445338066036013840064, −6.18176756141014332563203683225, −5.86375141046330158503967921225, −4.85666281770811325164981559373, −4.20373462910854671649122350459, −3.44411546708491957562188775606, −2.74717556756050260173470424649, −1.70791799525923622734495665147, −0.57499884741130546717190939202,
0.57499884741130546717190939202, 1.70791799525923622734495665147, 2.74717556756050260173470424649, 3.44411546708491957562188775606, 4.20373462910854671649122350459, 4.85666281770811325164981559373, 5.86375141046330158503967921225, 6.18176756141014332563203683225, 7.14771414445338066036013840064, 7.901636505504726285489337479754
