| L(s) = 1 | − 5-s + 7-s − 4·11-s − 6·13-s + 4·17-s + 6·19-s + 25-s + 6·29-s + 4·31-s − 35-s − 8·37-s + 10·41-s − 2·43-s − 10·47-s + 49-s − 14·53-s + 4·55-s − 4·59-s + 8·61-s + 6·65-s + 6·67-s + 2·71-s − 10·73-s − 4·77-s − 16·79-s − 8·83-s − 4·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s − 1.20·11-s − 1.66·13-s + 0.970·17-s + 1.37·19-s + 1/5·25-s + 1.11·29-s + 0.718·31-s − 0.169·35-s − 1.31·37-s + 1.56·41-s − 0.304·43-s − 1.45·47-s + 1/7·49-s − 1.92·53-s + 0.539·55-s − 0.520·59-s + 1.02·61-s + 0.744·65-s + 0.733·67-s + 0.237·71-s − 1.17·73-s − 0.455·77-s − 1.80·79-s − 0.878·83-s − 0.433·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−15.76920010375230, −15.64894996165811, −14.68443124828324, −14.31058491094600, −13.98395189351787, −13.03917156646324, −12.62353054782510, −12.05425569640069, −11.63590604399281, −11.02447694131759, −10.10055096344246, −10.05696423655043, −9.359029522656525, −8.396509944003596, −8.008099586790641, −7.413890325292358, −7.113147943064473, −6.098105001264112, −5.325409198104659, −4.923440885826325, −4.431047155875098, −3.170109490819733, −2.986274675527458, −2.031280160018517, −0.9977824864218143, 0,
0.9977824864218143, 2.031280160018517, 2.986274675527458, 3.170109490819733, 4.431047155875098, 4.923440885826325, 5.325409198104659, 6.098105001264112, 7.113147943064473, 7.413890325292358, 8.008099586790641, 8.396509944003596, 9.359029522656525, 10.05696423655043, 10.10055096344246, 11.02447694131759, 11.63590604399281, 12.05425569640069, 12.62353054782510, 13.03917156646324, 13.98395189351787, 14.31058491094600, 14.68443124828324, 15.64894996165811, 15.76920010375230
