| L(s) = 1 | − 5-s + 4·7-s + 4·11-s − 4·13-s − 4·17-s + 4·19-s + 8·23-s + 25-s + 6·29-s + 4·31-s − 4·35-s − 2·37-s − 10·41-s − 43-s + 4·47-s + 9·49-s − 2·53-s − 4·55-s + 12·59-s + 4·65-s + 8·67-s − 12·71-s − 14·73-s + 16·77-s − 8·79-s − 6·83-s + 4·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.51·7-s + 1.20·11-s − 1.10·13-s − 0.970·17-s + 0.917·19-s + 1.66·23-s + 1/5·25-s + 1.11·29-s + 0.718·31-s − 0.676·35-s − 0.328·37-s − 1.56·41-s − 0.152·43-s + 0.583·47-s + 9/7·49-s − 0.274·53-s − 0.539·55-s + 1.56·59-s + 0.496·65-s + 0.977·67-s − 1.42·71-s − 1.63·73-s + 1.82·77-s − 0.900·79-s − 0.658·83-s + 0.433·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 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
−13.95020110550278, −13.28091222582538, −12.84879974112314, −11.97951760200205, −11.86047840149738, −11.49111413779956, −11.07051318023211, −10.30907754129396, −10.05057286235285, −9.222702165297719, −8.740395133098602, −8.566876464131020, −7.804081408652888, −7.359141996696487, −6.833664599095433, −6.557790546869168, −5.552286401303692, −5.031650569517285, −4.703070281850928, −4.244093224750008, −3.543350411734880, −2.801245685149560, −2.301771749236656, −1.306308159066195, −1.161405240251585, 0,
1.161405240251585, 1.306308159066195, 2.301771749236656, 2.801245685149560, 3.543350411734880, 4.244093224750008, 4.703070281850928, 5.031650569517285, 5.552286401303692, 6.557790546869168, 6.833664599095433, 7.359141996696487, 7.804081408652888, 8.566876464131020, 8.740395133098602, 9.222702165297719, 10.05057286235285, 10.30907754129396, 11.07051318023211, 11.49111413779956, 11.86047840149738, 11.97951760200205, 12.84879974112314, 13.28091222582538, 13.95020110550278
