| L(s) = 1 | − 2.73·5-s + 4.46·7-s − 2.73·11-s + 13-s + 4.19·17-s − 4.46·19-s − 4.19·23-s + 2.46·25-s − 9.46·29-s − 3.46·31-s − 12.1·35-s + 2.46·37-s − 9.46·41-s + 3.46·43-s + 5.26·47-s + 12.9·49-s + 6.92·53-s + 7.46·55-s + 0.196·59-s + 13.9·61-s − 2.73·65-s − 11.9·67-s + 16.3·71-s − 13.9·73-s − 12.1·77-s − 79-s + 6.92·83-s + ⋯ |
| L(s) = 1 | − 1.22·5-s + 1.68·7-s − 0.823·11-s + 0.277·13-s + 1.01·17-s − 1.02·19-s − 0.874·23-s + 0.492·25-s − 1.75·29-s − 0.622·31-s − 2.06·35-s + 0.405·37-s − 1.47·41-s + 0.528·43-s + 0.768·47-s + 1.84·49-s + 0.951·53-s + 1.00·55-s + 0.0255·59-s + 1.78·61-s − 0.338·65-s − 1.45·67-s + 1.94·71-s − 1.63·73-s − 1.38·77-s − 0.112·79-s + 0.760·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.73T + 5T^{2} \) |
| 7 | \( 1 - 4.46T + 7T^{2} \) |
| 11 | \( 1 + 2.73T + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 - 4.19T + 17T^{2} \) |
| 19 | \( 1 + 4.46T + 19T^{2} \) |
| 23 | \( 1 + 4.19T + 23T^{2} \) |
| 29 | \( 1 + 9.46T + 29T^{2} \) |
| 31 | \( 1 + 3.46T + 31T^{2} \) |
| 37 | \( 1 - 2.46T + 37T^{2} \) |
| 41 | \( 1 + 9.46T + 41T^{2} \) |
| 43 | \( 1 - 3.46T + 43T^{2} \) |
| 47 | \( 1 - 5.26T + 47T^{2} \) |
| 53 | \( 1 - 6.92T + 53T^{2} \) |
| 59 | \( 1 - 0.196T + 59T^{2} \) |
| 61 | \( 1 - 13.9T + 61T^{2} \) |
| 67 | \( 1 + 11.9T + 67T^{2} \) |
| 71 | \( 1 - 16.3T + 71T^{2} \) |
| 73 | \( 1 + 13.9T + 73T^{2} \) |
| 79 | \( 1 + T + 79T^{2} \) |
| 83 | \( 1 - 6.92T + 83T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 + 16.3T + 97T^{2} \) |
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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
−8.158713740884210233433307581326, −7.67442706863661278834600213771, −7.04902126351868661818945273008, −5.67248569516435509421399991722, −5.23724902772298816066092544215, −4.16105203882009350791310416501, −3.81318093431089371995597246242, −2.44656205423574115741783141529, −1.46924812014072267460876671717, 0,
1.46924812014072267460876671717, 2.44656205423574115741783141529, 3.81318093431089371995597246242, 4.16105203882009350791310416501, 5.23724902772298816066092544215, 5.67248569516435509421399991722, 7.04902126351868661818945273008, 7.67442706863661278834600213771, 8.158713740884210233433307581326
