| L(s) = 1 | + 4-s − 2·9-s − 16·11-s + 4·16-s + 4·19-s + 8·29-s + 10·31-s − 2·36-s − 30·41-s − 16·44-s + 13·49-s + 6·59-s + 2·61-s + 11·64-s + 18·71-s + 4·76-s − 20·79-s + 3·81-s + 4·89-s + 32·99-s + 24·101-s − 36·109-s + 8·116-s + 126·121-s + 10·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 2/3·9-s − 4.82·11-s + 16-s + 0.917·19-s + 1.48·29-s + 1.79·31-s − 1/3·36-s − 4.68·41-s − 2.41·44-s + 13/7·49-s + 0.781·59-s + 0.256·61-s + 11/8·64-s + 2.13·71-s + 0.458·76-s − 2.25·79-s + 1/3·81-s + 0.423·89-s + 3.21·99-s + 2.38·101-s − 3.44·109-s + 0.742·116-s + 11.4·121-s + 0.898·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9814483267\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9814483267\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 - T^{2} - 3 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 13 T^{2} + 109 T^{4} - 13 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 8 T + 3 p T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 17 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 25 T^{2} + 633 T^{4} - 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 - 4 T + 57 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 5 T + 37 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 49 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 15 T + 127 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 145 T^{2} + 8893 T^{4} - 145 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 70 T^{2} + 3643 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 197 T^{2} + 15289 T^{4} - 197 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_4$ | \( ( 1 - 3 T - 31 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_4$ | \( ( 1 - T + 91 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 9 T + 151 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 43 T^{2} + 4089 T^{4} + 43 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 10 T + 178 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 209 T^{2} + 24637 T^{4} - 209 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 2 T + 159 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 185 T^{2} + 27273 T^{4} - 185 p^{2} T^{6} + p^{4} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.184914313639160075781557582664, −8.048992203311119554993948482167, −8.033489904254586694243826413676, −7.62510335338059115906055693201, −7.17129867272584766659044160619, −7.07345285312415941972646366686, −6.78093385471798345643467189250, −6.64298613900243841340829774688, −6.13840970801601055849563194524, −5.83702680614575928148881721426, −5.44793070893091285586227159966, −5.34436098192352677636921996920, −5.34271061012013849433052554493, −4.98426828630885020189546190149, −4.74089049241485198751686215205, −4.39796620724941943687918519115, −3.87074953732194923329507354264, −3.23434246316775919176751009167, −3.19080851116727452408226245903, −3.00104358916607971518895108346, −2.60651805165480462728901073425, −2.23511147531799918772124412750, −2.10509816619017424405030812614, −1.14484904531939549970433299779, −0.41033307837234712293814510122,
0.41033307837234712293814510122, 1.14484904531939549970433299779, 2.10509816619017424405030812614, 2.23511147531799918772124412750, 2.60651805165480462728901073425, 3.00104358916607971518895108346, 3.19080851116727452408226245903, 3.23434246316775919176751009167, 3.87074953732194923329507354264, 4.39796620724941943687918519115, 4.74089049241485198751686215205, 4.98426828630885020189546190149, 5.34271061012013849433052554493, 5.34436098192352677636921996920, 5.44793070893091285586227159966, 5.83702680614575928148881721426, 6.13840970801601055849563194524, 6.64298613900243841340829774688, 6.78093385471798345643467189250, 7.07345285312415941972646366686, 7.17129867272584766659044160619, 7.62510335338059115906055693201, 8.033489904254586694243826413676, 8.048992203311119554993948482167, 8.184914313639160075781557582664
