| L(s) = 1 | + 2·4-s − 5·16-s − 10·25-s − 8·43-s − 4·49-s − 28·61-s − 20·64-s + 32·79-s − 20·100-s − 8·103-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 16·172-s + 173-s + 179-s + 181-s + 191-s + 193-s − 8·196-s + ⋯ |
| L(s) = 1 | + 4-s − 5/4·16-s − 2·25-s − 1.21·43-s − 4/7·49-s − 3.58·61-s − 5/2·64-s + 3.60·79-s − 2·100-s − 0.788·103-s − 0.363·121-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 + 0.0783·163-s + 0.0773·167-s − 1.21·172-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s − 4/7·196-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.987488796\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.987488796\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 19 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 43 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 71 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 77 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 29 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 97 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 146 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 158 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 146 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.63339800447605068874570904626, −6.57650722137274033273145283721, −6.54000410023318868922375130505, −6.11935604460953131572433872004, −6.08368432190686119347373147348, −5.58706080568863421411953848254, −5.39951484836429333868404903685, −5.26871767392059862585718345662, −5.23316109809968390773540466347, −4.56188853449087200507201522328, −4.43922427981244934972207506445, −4.35924213581238394881552865739, −4.22225059875039158251176681207, −3.87035595160793097875695030727, −3.33935580569889001720223598412, −3.18262028694788129275737234824, −3.12247599941473213509210880389, −2.92602096648986969988191730383, −2.28831522888973962679995276751, −2.12319073653516606276439508814, −1.93693266734966616326600412945, −1.75473925637323638732848827979, −1.42111218525009464563289843159, −0.70482319783376986506171929457, −0.29872830869381274131204375417,
0.29872830869381274131204375417, 0.70482319783376986506171929457, 1.42111218525009464563289843159, 1.75473925637323638732848827979, 1.93693266734966616326600412945, 2.12319073653516606276439508814, 2.28831522888973962679995276751, 2.92602096648986969988191730383, 3.12247599941473213509210880389, 3.18262028694788129275737234824, 3.33935580569889001720223598412, 3.87035595160793097875695030727, 4.22225059875039158251176681207, 4.35924213581238394881552865739, 4.43922427981244934972207506445, 4.56188853449087200507201522328, 5.23316109809968390773540466347, 5.26871767392059862585718345662, 5.39951484836429333868404903685, 5.58706080568863421411953848254, 6.08368432190686119347373147348, 6.11935604460953131572433872004, 6.54000410023318868922375130505, 6.57650722137274033273145283721, 6.63339800447605068874570904626
