| L(s) = 1 | − 2·5-s + 7-s − 5·11-s + 5·13-s − 4·17-s + 2·19-s + 7·23-s + 3·25-s − 5·29-s + 5·31-s − 2·35-s − 6·37-s + 8·43-s − 5·47-s + 49-s − 9·53-s + 10·55-s + 26·59-s + 2·61-s − 10·65-s − 14·67-s + 9·71-s − 5·77-s + 6·79-s − 10·83-s + 8·85-s − 3·89-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.377·7-s − 1.50·11-s + 1.38·13-s − 0.970·17-s + 0.458·19-s + 1.45·23-s + 3/5·25-s − 0.928·29-s + 0.898·31-s − 0.338·35-s − 0.986·37-s + 1.21·43-s − 0.729·47-s + 1/7·49-s − 1.23·53-s + 1.34·55-s + 3.38·59-s + 0.256·61-s − 1.24·65-s − 1.71·67-s + 1.06·71-s − 0.569·77-s + 0.675·79-s − 1.09·83-s + 0.867·85-s − 0.317·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.843352830\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.843352830\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - T - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 5 T + 18 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 7 T + 44 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 5 T + 50 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 5 T + 54 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 5 T + 86 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 9 T + 112 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 2 T + 66 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 14 T + 126 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 9 T + 148 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 6 T + 110 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 10 T + 134 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 3 T + 52 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.650252902838526726958790535983, −8.597384557843943529786813673015, −7.939727125040869020155615769110, −7.913217828865460142882300000029, −7.36933722094255852270549795771, −7.05960089397053320680793935301, −6.64423680773547303557952435588, −6.31209774848475950501927613175, −5.67184343606088208260550298262, −5.37557109151116806806331633957, −5.06188546021354026268271835300, −4.59101359525758670755932294012, −4.16850117286836626294597103096, −3.78457887691571518364894280035, −3.13152140550341023343367265983, −3.04534604546798916282346167792, −2.30453229299001038824384109961, −1.83263659250988989726856207869, −1.02789868228711771186798810681, −0.49072456649946162630146174236,
0.49072456649946162630146174236, 1.02789868228711771186798810681, 1.83263659250988989726856207869, 2.30453229299001038824384109961, 3.04534604546798916282346167792, 3.13152140550341023343367265983, 3.78457887691571518364894280035, 4.16850117286836626294597103096, 4.59101359525758670755932294012, 5.06188546021354026268271835300, 5.37557109151116806806331633957, 5.67184343606088208260550298262, 6.31209774848475950501927613175, 6.64423680773547303557952435588, 7.05960089397053320680793935301, 7.36933722094255852270549795771, 7.913217828865460142882300000029, 7.939727125040869020155615769110, 8.597384557843943529786813673015, 8.650252902838526726958790535983
