| L(s) = 1 | − 2·5-s + 4·11-s + 24·19-s + 5·25-s − 18·29-s − 4·31-s − 22·41-s − 13·49-s − 8·55-s − 8·59-s + 14·61-s − 24·71-s + 24·79-s + 4·89-s − 48·95-s + 4·101-s − 28·109-s + 26·121-s − 22·125-s + 127-s + 131-s + 137-s + 139-s + 36·145-s + 149-s + 151-s + 8·155-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.20·11-s + 5.50·19-s + 25-s − 3.34·29-s − 0.718·31-s − 3.43·41-s − 1.85·49-s − 1.07·55-s − 1.04·59-s + 1.79·61-s − 2.84·71-s + 2.70·79-s + 0.423·89-s − 4.92·95-s + 0.398·101-s − 2.68·109-s + 2.36·121-s − 1.96·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.98·145-s + 0.0819·149-s + 0.0813·151-s + 0.642·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.220270998\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.220270998\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| good | 7 | $C_2^2$$\times$$C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )( 1 + 11 T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^2$ | \( ( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 + 45 T^{2} + 1496 T^{4} + 45 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 11 T + 80 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 + 70 T^{2} + 3051 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^3$ | \( 1 + 45 T^{2} - 184 T^{4} + 45 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 109 T^{2} + p^{2} T^{4} )( 1 + 122 T^{2} + p^{2} T^{4} ) \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 12 T + 65 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 45 T^{2} - 4864 T^{4} + 45 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 97 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 18 T + 227 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 227 T^{2} + 18 p T^{3} + p^{2} T^{4} ) \) |
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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
−6.39191507468739030506084472433, −6.27898335818023946931470861574, −6.12250079706366030990318680996, −5.69356978117290734652475038992, −5.49056207877972084821920575548, −5.32702715448864598836607043906, −5.14358854560301000579086809992, −5.11476058560992802893040548421, −4.91160904727196451621835533681, −4.62239881318503279232934133152, −4.22492702646077614777953787587, −3.81961568536593486504635124916, −3.78681684856558431867977461428, −3.58862377387793495029572050168, −3.41854288121771279328849632459, −3.26847093108326051353512080636, −3.05142900184238625552501790323, −2.72345703385076259328850703366, −2.43713061753857832029261631720, −1.77013564907901542014251415948, −1.71475368722205038123219230891, −1.35567415451628739427814281445, −1.28178769109159176735238708802, −0.790075092392969020898574693832, −0.20095382087468369658917649633,
0.20095382087468369658917649633, 0.790075092392969020898574693832, 1.28178769109159176735238708802, 1.35567415451628739427814281445, 1.71475368722205038123219230891, 1.77013564907901542014251415948, 2.43713061753857832029261631720, 2.72345703385076259328850703366, 3.05142900184238625552501790323, 3.26847093108326051353512080636, 3.41854288121771279328849632459, 3.58862377387793495029572050168, 3.78681684856558431867977461428, 3.81961568536593486504635124916, 4.22492702646077614777953787587, 4.62239881318503279232934133152, 4.91160904727196451621835533681, 5.11476058560992802893040548421, 5.14358854560301000579086809992, 5.32702715448864598836607043906, 5.49056207877972084821920575548, 5.69356978117290734652475038992, 6.12250079706366030990318680996, 6.27898335818023946931470861574, 6.39191507468739030506084472433
