| L(s) = 1 | + 8·5-s − 12·13-s + 24·17-s + 32·25-s + 16·29-s − 12·37-s − 8·49-s + 16·53-s − 12·61-s − 96·65-s + 192·85-s − 16·97-s + 32·101-s + 28·109-s − 24·113-s + 104·125-s + 127-s + 131-s + 137-s + 139-s + 128·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 72·169-s + ⋯ |
| L(s) = 1 | + 3.57·5-s − 3.32·13-s + 5.82·17-s + 32/5·25-s + 2.97·29-s − 1.97·37-s − 8/7·49-s + 2.19·53-s − 1.53·61-s − 11.9·65-s + 20.8·85-s − 1.62·97-s + 3.18·101-s + 2.68·109-s − 2.25·113-s + 9.30·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 10.6·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 5.53·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(14.96746368\) |
| \(L(\frac12)\) |
\(\approx\) |
\(14.96746368\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - 206 T^{4} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} ) \) |
| 23 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 44 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2^3$ | \( 1 - 1198 T^{4} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 4946 T^{4} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 140 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 5678 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 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
−6.40674410564623498435249081113, −5.98844311392700331368739220852, −5.85994842223965512231748820005, −5.69575739591122151734005246437, −5.56274248591002530826125794123, −5.50820553313430614408276083830, −5.24520231835866757431053605976, −4.89216196036654126963001983910, −4.88400112366028489997712661517, −4.80397716659773166444022497642, −4.54417497817803933031489401397, −3.89394196296759905334504483012, −3.85012808496995965918658611942, −3.27585321719387904688766880416, −3.05326194989636563524044788726, −3.04684825007837185765112680077, −3.01101206757147578648446175860, −2.40815721792198447247373615429, −2.38267782007280543547009143943, −2.04690827191112336947048231876, −1.75056685549526493265060082469, −1.40427380150709814908647425834, −1.32938855879292541447927399206, −0.808573877999497436620348731660, −0.61146414010691694521683504386,
0.61146414010691694521683504386, 0.808573877999497436620348731660, 1.32938855879292541447927399206, 1.40427380150709814908647425834, 1.75056685549526493265060082469, 2.04690827191112336947048231876, 2.38267782007280543547009143943, 2.40815721792198447247373615429, 3.01101206757147578648446175860, 3.04684825007837185765112680077, 3.05326194989636563524044788726, 3.27585321719387904688766880416, 3.85012808496995965918658611942, 3.89394196296759905334504483012, 4.54417497817803933031489401397, 4.80397716659773166444022497642, 4.88400112366028489997712661517, 4.89216196036654126963001983910, 5.24520231835866757431053605976, 5.50820553313430614408276083830, 5.56274248591002530826125794123, 5.69575739591122151734005246437, 5.85994842223965512231748820005, 5.98844311392700331368739220852, 6.40674410564623498435249081113
