| L(s) = 1 | + 5-s − 2·7-s − 2·13-s − 6·17-s + 4·19-s − 6·23-s + 25-s + 6·29-s − 4·31-s − 2·35-s + 2·37-s + 6·41-s + 10·43-s + 6·47-s − 3·49-s + 6·53-s − 12·59-s − 2·61-s − 2·65-s + 2·67-s + 12·71-s − 2·73-s − 8·79-s + 6·83-s − 6·85-s + 6·89-s + 4·91-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s − 0.554·13-s − 1.45·17-s + 0.917·19-s − 1.25·23-s + 1/5·25-s + 1.11·29-s − 0.718·31-s − 0.338·35-s + 0.328·37-s + 0.937·41-s + 1.52·43-s + 0.875·47-s − 3/7·49-s + 0.824·53-s − 1.56·59-s − 0.256·61-s − 0.248·65-s + 0.244·67-s + 1.42·71-s − 0.234·73-s − 0.900·79-s + 0.658·83-s − 0.650·85-s + 0.635·89-s + 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−15.91053990007987, −15.42718898683764, −14.68137790561565, −13.97886570928675, −13.83554966501341, −13.09190740342869, −12.52885682591703, −12.18772160061789, −11.38745522362678, −10.88390056246560, −10.19173935643563, −9.781093817554055, −9.108607155862283, −8.844836097533042, −7.802350900167554, −7.434532923981769, −6.621864696153497, −6.184018143160976, −5.610898372444905, −4.783000451526619, −4.212513262415596, −3.449131971724261, −2.566602231436570, −2.179483921647313, −1.010923854357220, 0,
1.010923854357220, 2.179483921647313, 2.566602231436570, 3.449131971724261, 4.212513262415596, 4.783000451526619, 5.610898372444905, 6.184018143160976, 6.621864696153497, 7.434532923981769, 7.802350900167554, 8.844836097533042, 9.108607155862283, 9.781093817554055, 10.19173935643563, 10.88390056246560, 11.38745522362678, 12.18772160061789, 12.52885682591703, 13.09190740342869, 13.83554966501341, 13.97886570928675, 14.68137790561565, 15.42718898683764, 15.91053990007987
