| L(s) = 1 | − 3.46·5-s − 2.73·7-s − 1.26·11-s − 5.46·13-s + 17-s + 1.46·19-s + 1.26·23-s + 6.99·25-s + 3.46·29-s + 4.19·31-s + 9.46·35-s − 4.53·37-s + 6·41-s + 8.39·43-s + 6.92·47-s + 0.464·49-s + 12.9·53-s + 4.39·55-s + 2.53·59-s + 0.535·61-s + 18.9·65-s − 14.9·67-s + 8.19·71-s + 2·73-s + 3.46·77-s − 12.1·79-s − 2.53·83-s + ⋯ |
| L(s) = 1 | − 1.54·5-s − 1.03·7-s − 0.382·11-s − 1.51·13-s + 0.242·17-s + 0.335·19-s + 0.264·23-s + 1.39·25-s + 0.643·29-s + 0.753·31-s + 1.59·35-s − 0.745·37-s + 0.937·41-s + 1.27·43-s + 1.01·47-s + 0.0663·49-s + 1.77·53-s + 0.592·55-s + 0.330·59-s + 0.0686·61-s + 2.34·65-s − 1.82·67-s + 0.972·71-s + 0.234·73-s + 0.394·77-s − 1.37·79-s − 0.278·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 3.46T + 5T^{2} \) |
| 7 | \( 1 + 2.73T + 7T^{2} \) |
| 11 | \( 1 + 1.26T + 11T^{2} \) |
| 13 | \( 1 + 5.46T + 13T^{2} \) |
| 19 | \( 1 - 1.46T + 19T^{2} \) |
| 23 | \( 1 - 1.26T + 23T^{2} \) |
| 29 | \( 1 - 3.46T + 29T^{2} \) |
| 31 | \( 1 - 4.19T + 31T^{2} \) |
| 37 | \( 1 + 4.53T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 8.39T + 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 - 2.53T + 59T^{2} \) |
| 61 | \( 1 - 0.535T + 61T^{2} \) |
| 67 | \( 1 + 14.9T + 67T^{2} \) |
| 71 | \( 1 - 8.19T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 + 2.53T + 83T^{2} \) |
| 89 | \( 1 + 2.53T + 89T^{2} \) |
| 97 | \( 1 + 4.92T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.28002282101717724230790324787, −6.97854172489122686917638755212, −5.98126657208506740057739226688, −5.19028251777974610554366868457, −4.42198203211738445711952393374, −3.86448379631288230893354718176, −2.96207482245245478389092757692, −2.54295528130151363942129121755, −0.850133744635735652625027308449, 0,
0.850133744635735652625027308449, 2.54295528130151363942129121755, 2.96207482245245478389092757692, 3.86448379631288230893354718176, 4.42198203211738445711952393374, 5.19028251777974610554366868457, 5.98126657208506740057739226688, 6.97854172489122686917638755212, 7.28002282101717724230790324787
