| L(s) = 1 | + 9·5-s + 31·7-s − 15·11-s − 37·13-s + 42·17-s + 28·19-s + 195·23-s − 44·25-s − 111·29-s + 205·31-s + 279·35-s − 166·37-s + 261·41-s + 43·43-s + 177·47-s + 618·49-s − 114·53-s − 135·55-s + 159·59-s + 191·61-s − 333·65-s + 421·67-s + 156·71-s + 182·73-s − 465·77-s − 1.13e3·79-s − 1.08e3·83-s + ⋯ |
| L(s) = 1 | + 0.804·5-s + 1.67·7-s − 0.411·11-s − 0.789·13-s + 0.599·17-s + 0.338·19-s + 1.76·23-s − 0.351·25-s − 0.710·29-s + 1.18·31-s + 1.34·35-s − 0.737·37-s + 0.994·41-s + 0.152·43-s + 0.549·47-s + 1.80·49-s − 0.295·53-s − 0.330·55-s + 0.350·59-s + 0.400·61-s − 0.635·65-s + 0.767·67-s + 0.260·71-s + 0.291·73-s − 0.688·77-s − 1.61·79-s − 1.43·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.261229088\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.261229088\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 9 T + p^{3} T^{2} \) |
| 7 | \( 1 - 31 T + p^{3} T^{2} \) |
| 11 | \( 1 + 15 T + p^{3} T^{2} \) |
| 13 | \( 1 + 37 T + p^{3} T^{2} \) |
| 17 | \( 1 - 42 T + p^{3} T^{2} \) |
| 19 | \( 1 - 28 T + p^{3} T^{2} \) |
| 23 | \( 1 - 195 T + p^{3} T^{2} \) |
| 29 | \( 1 + 111 T + p^{3} T^{2} \) |
| 31 | \( 1 - 205 T + p^{3} T^{2} \) |
| 37 | \( 1 + 166 T + p^{3} T^{2} \) |
| 41 | \( 1 - 261 T + p^{3} T^{2} \) |
| 43 | \( 1 - p T + p^{3} T^{2} \) |
| 47 | \( 1 - 177 T + p^{3} T^{2} \) |
| 53 | \( 1 + 114 T + p^{3} T^{2} \) |
| 59 | \( 1 - 159 T + p^{3} T^{2} \) |
| 61 | \( 1 - 191 T + p^{3} T^{2} \) |
| 67 | \( 1 - 421 T + p^{3} T^{2} \) |
| 71 | \( 1 - 156 T + p^{3} T^{2} \) |
| 73 | \( 1 - 182 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1133 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1083 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1050 T + p^{3} T^{2} \) |
| 97 | \( 1 + 901 T + p^{3} T^{2} \) |
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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
−9.320273173894260984714815690609, −8.424815599384315930766334221184, −7.67759974536129000334768677359, −6.98205811420524123855911819601, −5.63574592115388834166004270235, −5.19240098143359732203485962259, −4.35613785306754757519506359359, −2.87344381244706326024240919920, −1.94269867512205913631379968250, −0.968643922886922191920894929279,
0.968643922886922191920894929279, 1.94269867512205913631379968250, 2.87344381244706326024240919920, 4.35613785306754757519506359359, 5.19240098143359732203485962259, 5.63574592115388834166004270235, 6.98205811420524123855911819601, 7.67759974536129000334768677359, 8.424815599384315930766334221184, 9.320273173894260984714815690609
