| L(s) = 1 | − 7-s + 4·11-s + 6·13-s + 6·17-s − 4·19-s − 4·23-s + 2·29-s + 8·31-s − 6·37-s − 6·41-s − 8·43-s + 49-s + 6·53-s + 4·59-s + 10·61-s − 8·67-s − 12·71-s + 14·73-s − 4·77-s − 16·79-s − 12·83-s − 14·89-s − 6·91-s − 18·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 1.20·11-s + 1.66·13-s + 1.45·17-s − 0.917·19-s − 0.834·23-s + 0.371·29-s + 1.43·31-s − 0.986·37-s − 0.937·41-s − 1.21·43-s + 1/7·49-s + 0.824·53-s + 0.520·59-s + 1.28·61-s − 0.977·67-s − 1.42·71-s + 1.63·73-s − 0.455·77-s − 1.80·79-s − 1.31·83-s − 1.48·89-s − 0.628·91-s − 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−14.75689936661782, −14.08709624831641, −13.87838033236037, −13.29062253018191, −12.72205996981142, −12.09735013182398, −11.75405579840769, −11.30974379968170, −10.51580246614329, −10.03418001473456, −9.803575163418833, −8.811246383852347, −8.509225083937606, −8.206072300389117, −7.260651389187452, −6.660953820159949, −6.307170731901660, −5.767745727858121, −5.144957844516117, −4.167262401316608, −3.866736133319649, −3.300922189363952, −2.553891848750251, −1.398349045162365, −1.267311739090563, 0,
1.267311739090563, 1.398349045162365, 2.553891848750251, 3.300922189363952, 3.866736133319649, 4.167262401316608, 5.144957844516117, 5.767745727858121, 6.307170731901660, 6.660953820159949, 7.260651389187452, 8.206072300389117, 8.509225083937606, 8.811246383852347, 9.803575163418833, 10.03418001473456, 10.51580246614329, 11.30974379968170, 11.75405579840769, 12.09735013182398, 12.72205996981142, 13.29062253018191, 13.87838033236037, 14.08709624831641, 14.75689936661782
