| L(s) = 1 | + 3·5-s − 6·7-s + 6·11-s − 6·17-s + 5·25-s − 18·35-s + 24·43-s + 11·49-s + 12·53-s + 18·55-s + 12·59-s + 22·61-s + 36·67-s − 36·77-s − 18·85-s + 22·109-s + 12·113-s + 36·119-s − 5·121-s + 18·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 2.26·7-s + 1.80·11-s − 1.45·17-s + 25-s − 3.04·35-s + 3.65·43-s + 11/7·49-s + 1.64·53-s + 2.42·55-s + 1.56·59-s + 2.81·61-s + 4.39·67-s − 4.10·77-s − 1.95·85-s + 2.10·109-s + 1.12·113-s + 3.30·119-s − 0.454·121-s + 1.60·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-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
| \(L(1)\) |
\(\approx\) |
\(4.358895870\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.358895870\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| good | 7 | $D_{4}$ | \( ( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_{4}$ | \( ( 1 - 3 T + 16 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 + 3 T + 28 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 25 T^{2} + 804 T^{4} - 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 41 T^{2} + 1404 T^{4} - 41 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 40 T^{2} + 894 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 88 T^{2} + 4110 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 59 | $D_{4}$ | \( ( 1 - 6 T + 94 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 11 T + 78 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 18 T + 182 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 155 T^{2} + 15996 T^{4} + 155 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 253 T^{2} + 27816 T^{4} - 253 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 110 T^{2} + 12051 T^{4} - 110 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 217 T^{2} + 24576 T^{4} - 217 p^{2} T^{6} + p^{4} T^{8} \) |
| 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
−6.59591989853160728361229085380, −6.14637752831145232464239941634, −6.00857270864712435662944824383, −5.96838985078712424839010301244, −5.78389124382186022751775724027, −5.52621889583253606394902536536, −5.13856755053604735724467175692, −5.07630565758247074950041596431, −4.85175486693120023857803267377, −4.44502068325708001735722729994, −4.10399929444701780770986322229, −3.99865346295344102090686601581, −3.91796271545713828688548292204, −3.68761963389447575571679360624, −3.37808282224618244030100608307, −3.14777086397621828075512634365, −2.89006660164479664498895725427, −2.31629908635367358053328557761, −2.22704655257250943678137057552, −2.21235064956171672515243806855, −2.19098813725216934666578866754, −1.28072977052046035449416812296, −1.00283764021904328094958535124, −0.854037541576172952218092913356, −0.39474973790131639170915055398,
0.39474973790131639170915055398, 0.854037541576172952218092913356, 1.00283764021904328094958535124, 1.28072977052046035449416812296, 2.19098813725216934666578866754, 2.21235064956171672515243806855, 2.22704655257250943678137057552, 2.31629908635367358053328557761, 2.89006660164479664498895725427, 3.14777086397621828075512634365, 3.37808282224618244030100608307, 3.68761963389447575571679360624, 3.91796271545713828688548292204, 3.99865346295344102090686601581, 4.10399929444701780770986322229, 4.44502068325708001735722729994, 4.85175486693120023857803267377, 5.07630565758247074950041596431, 5.13856755053604735724467175692, 5.52621889583253606394902536536, 5.78389124382186022751775724027, 5.96838985078712424839010301244, 6.00857270864712435662944824383, 6.14637752831145232464239941634, 6.59591989853160728361229085380
