| L(s) = 1 | + 3.60·5-s + 3.60·7-s + 11-s + 4·13-s − 7.21·19-s − 6·23-s + 7.99·25-s − 7.21·29-s − 3.60·31-s + 12.9·35-s + 10·37-s − 7.21·41-s + 7.21·43-s − 10·47-s + 5.99·49-s − 3.60·53-s + 3.60·55-s − 4·59-s + 14.4·65-s + 7.21·67-s + 8·71-s − 3·73-s + 3.60·77-s − 14.4·79-s + 9·83-s + 7.21·89-s + 14.4·91-s + ⋯ |
| L(s) = 1 | + 1.61·5-s + 1.36·7-s + 0.301·11-s + 1.10·13-s − 1.65·19-s − 1.25·23-s + 1.59·25-s − 1.33·29-s − 0.647·31-s + 2.19·35-s + 1.64·37-s − 1.12·41-s + 1.09·43-s − 1.45·47-s + 0.857·49-s − 0.495·53-s + 0.486·55-s − 0.520·59-s + 1.78·65-s + 0.880·67-s + 0.949·71-s − 0.351·73-s + 0.410·77-s − 1.62·79-s + 0.987·83-s + 0.764·89-s + 1.51·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.382052951\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.382052951\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.60T + 5T^{2} \) |
| 7 | \( 1 - 3.60T + 7T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 7.21T + 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + 7.21T + 29T^{2} \) |
| 31 | \( 1 + 3.60T + 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 + 7.21T + 41T^{2} \) |
| 43 | \( 1 - 7.21T + 43T^{2} \) |
| 47 | \( 1 + 10T + 47T^{2} \) |
| 53 | \( 1 + 3.60T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 7.21T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 3T + 73T^{2} \) |
| 79 | \( 1 + 14.4T + 79T^{2} \) |
| 83 | \( 1 - 9T + 83T^{2} \) |
| 89 | \( 1 - 7.21T + 89T^{2} \) |
| 97 | \( 1 - 7T + 97T^{2} \) |
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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
−10.19534150273239744493597875300, −9.296526775619267065101785393675, −8.564140691251407862935919999075, −7.77288352678457506057851459326, −6.37738817600451278796826203381, −5.93676176140044070660730530222, −4.93288783143059428149116711677, −3.90339755045674366825414658449, −2.16630677037808489600931869686, −1.57532584710971668528531947934,
1.57532584710971668528531947934, 2.16630677037808489600931869686, 3.90339755045674366825414658449, 4.93288783143059428149116711677, 5.93676176140044070660730530222, 6.37738817600451278796826203381, 7.77288352678457506057851459326, 8.564140691251407862935919999075, 9.296526775619267065101785393675, 10.19534150273239744493597875300
