| L(s) = 1 | + 5-s + 2.73·7-s + 1.73·11-s + 5.46·13-s + 4.73·17-s + 4.46·19-s + 3.46·23-s + 25-s + 7.73·29-s − 5.92·31-s + 2.73·35-s − 6.19·37-s + 11.1·41-s − 3.26·43-s − 1.26·47-s + 0.464·49-s + 7.26·53-s + 1.73·55-s − 7.73·59-s − 4·61-s + 5.46·65-s − 6.39·67-s − 11.1·71-s − 0.196·73-s + 4.73·77-s + 14.3·79-s − 15.1·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.03·7-s + 0.522·11-s + 1.51·13-s + 1.14·17-s + 1.02·19-s + 0.722·23-s + 0.200·25-s + 1.43·29-s − 1.06·31-s + 0.461·35-s − 1.01·37-s + 1.74·41-s − 0.498·43-s − 0.184·47-s + 0.0663·49-s + 0.998·53-s + 0.233·55-s − 1.00·59-s − 0.512·61-s + 0.677·65-s − 0.780·67-s − 1.32·71-s − 0.0229·73-s + 0.539·77-s + 1.61·79-s − 1.66·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.356434447\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.356434447\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 7 | \( 1 - 2.73T + 7T^{2} \) |
| 11 | \( 1 - 1.73T + 11T^{2} \) |
| 13 | \( 1 - 5.46T + 13T^{2} \) |
| 17 | \( 1 - 4.73T + 17T^{2} \) |
| 19 | \( 1 - 4.46T + 19T^{2} \) |
| 23 | \( 1 - 3.46T + 23T^{2} \) |
| 29 | \( 1 - 7.73T + 29T^{2} \) |
| 31 | \( 1 + 5.92T + 31T^{2} \) |
| 37 | \( 1 + 6.19T + 37T^{2} \) |
| 41 | \( 1 - 11.1T + 41T^{2} \) |
| 43 | \( 1 + 3.26T + 43T^{2} \) |
| 47 | \( 1 + 1.26T + 47T^{2} \) |
| 53 | \( 1 - 7.26T + 53T^{2} \) |
| 59 | \( 1 + 7.73T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 + 6.39T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 + 0.196T + 73T^{2} \) |
| 79 | \( 1 - 14.3T + 79T^{2} \) |
| 83 | \( 1 + 15.1T + 83T^{2} \) |
| 89 | \( 1 + 5.19T + 89T^{2} \) |
| 97 | \( 1 - 0.732T + 97T^{2} \) |
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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
−8.022175519126736491804318345645, −7.38412984418221750287415715713, −6.57701673419487594083303580603, −5.77589532330326647914965185883, −5.29434939869546583564258902035, −4.43330748845971855041970699839, −3.57635828465849703022253879022, −2.83201397202957202783191456119, −1.47216118033436652737423148887, −1.16534587693512391321881678275,
1.16534587693512391321881678275, 1.47216118033436652737423148887, 2.83201397202957202783191456119, 3.57635828465849703022253879022, 4.43330748845971855041970699839, 5.29434939869546583564258902035, 5.77589532330326647914965185883, 6.57701673419487594083303580603, 7.38412984418221750287415715713, 8.022175519126736491804318345645
