| L(s) = 1 | − 2·3-s + 6·7-s + 2·9-s − 12·21-s − 6·27-s − 32·41-s + 20·43-s + 4·47-s + 18·49-s + 12·63-s − 12·67-s + 24·79-s + 11·81-s + 20·83-s − 32·89-s + 8·101-s + 24·109-s + 16·121-s + 64·123-s + 127-s − 40·129-s + 131-s + 137-s + 139-s − 8·141-s − 36·147-s + 149-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 2.26·7-s + 2/3·9-s − 2.61·21-s − 1.15·27-s − 4.99·41-s + 3.04·43-s + 0.583·47-s + 18/7·49-s + 1.51·63-s − 1.46·67-s + 2.70·79-s + 11/9·81-s + 2.19·83-s − 3.39·89-s + 0.796·101-s + 2.29·109-s + 1.45·121-s + 5.77·123-s + 0.0887·127-s − 3.52·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.673·141-s − 2.96·147-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.873897270\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.873897270\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| good | 11 | $C_4\times C_2$ | \( 1 - 16 T^{2} + 126 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 40 T^{2} + 718 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 286 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 112 T^{2} + 5038 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_4$ | \( ( 1 + 16 T + 126 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 10 T + 106 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 - 2 T + 90 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 104 T^{2} + 7342 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 2998 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 6 T + 98 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 32 T^{2} + 9838 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 120 T^{2} + 12638 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 12 T + 174 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 10 T + 186 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 16 T + 222 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 120 T^{2} + 17918 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.50330565594733152672448542381, −6.11825353391307875809058701712, −6.03915487913356727881254102953, −5.76125100715861922226080258357, −5.63814356745175157398282963903, −5.39205856383855464260407131328, −5.13633520390968627427637082393, −5.04082326892570329867920399680, −4.90378206557587266961397513436, −4.52618376836488856001687656231, −4.41155848360113745784466762464, −4.34775873897108425031676013372, −3.83647402218238611384733974009, −3.71651721174080239674426289323, −3.52642669839284384648282888434, −3.06128127042124562001491501700, −2.99269128725705143221973181487, −2.52933301558352467839479114322, −2.09012959192977059525731073856, −2.00370046655772203039439130270, −1.68156652271660221486212407791, −1.62221810851145744902809632069, −1.07276819059625396837706510844, −0.72075156110284115088837230103, −0.37522668597895561859101613654,
0.37522668597895561859101613654, 0.72075156110284115088837230103, 1.07276819059625396837706510844, 1.62221810851145744902809632069, 1.68156652271660221486212407791, 2.00370046655772203039439130270, 2.09012959192977059525731073856, 2.52933301558352467839479114322, 2.99269128725705143221973181487, 3.06128127042124562001491501700, 3.52642669839284384648282888434, 3.71651721174080239674426289323, 3.83647402218238611384733974009, 4.34775873897108425031676013372, 4.41155848360113745784466762464, 4.52618376836488856001687656231, 4.90378206557587266961397513436, 5.04082326892570329867920399680, 5.13633520390968627427637082393, 5.39205856383855464260407131328, 5.63814356745175157398282963903, 5.76125100715861922226080258357, 6.03915487913356727881254102953, 6.11825353391307875809058701712, 6.50330565594733152672448542381
