| L(s) = 1 | − 5·9-s + 4·11-s + 12·19-s + 12·29-s − 28·41-s + 13·49-s − 20·59-s − 10·61-s − 40·71-s + 12·79-s + 9·81-s + 34·89-s − 20·99-s + 34·101-s − 10·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 20·169-s − 60·171-s + ⋯ |
| L(s) = 1 | − 5/3·9-s + 1.20·11-s + 2.75·19-s + 2.22·29-s − 4.37·41-s + 13/7·49-s − 2.60·59-s − 1.28·61-s − 4.74·71-s + 1.35·79-s + 81-s + 3.60·89-s − 2.01·99-s + 3.38·101-s − 0.957·109-s + 2.36·121-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 + 1.53·169-s − 4.58·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1388873904\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1388873904\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| good | 3 | $C_2^3$ | \( 1 + 5 T^{2} + 16 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 37 T^{2} + 840 T^{4} + 37 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \) |
| 41 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^3$ | \( 1 + 70 T^{2} + 2091 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 10 T + 41 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 + 82 T^{2} + 1395 T^{4} + 82 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 6 T - 43 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 157 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 17 T + 200 T^{2} - 17 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 190 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.83078533784247317800508408900, −6.31936931835528091249048309640, −6.29147724460214324233297888925, −6.27758126451646121187334014346, −5.99768990619082366079163978303, −5.99713427961379273547716370704, −5.38545465149968097661784053724, −5.13014655968554989062592739967, −5.05394865355582098786243154960, −5.00970025529159122714410260969, −4.74458856265836579783423488414, −4.27460452263891181140785659261, −4.25683563651642515280401964899, −3.64342013336130122110598624939, −3.50118076867954482843762302973, −3.21114239033019530988827467853, −3.14883063946513827653123226655, −3.11786366517122804357721820476, −2.41873872865100778907975871267, −2.41415810248508784954108643651, −1.95411974131313597969562195462, −1.30801961132419506638461242743, −1.18993694856978543327212632836, −1.13915499897296286695030653829, −0.07322748617058085621290372614,
0.07322748617058085621290372614, 1.13915499897296286695030653829, 1.18993694856978543327212632836, 1.30801961132419506638461242743, 1.95411974131313597969562195462, 2.41415810248508784954108643651, 2.41873872865100778907975871267, 3.11786366517122804357721820476, 3.14883063946513827653123226655, 3.21114239033019530988827467853, 3.50118076867954482843762302973, 3.64342013336130122110598624939, 4.25683563651642515280401964899, 4.27460452263891181140785659261, 4.74458856265836579783423488414, 5.00970025529159122714410260969, 5.05394865355582098786243154960, 5.13014655968554989062592739967, 5.38545465149968097661784053724, 5.99713427961379273547716370704, 5.99768990619082366079163978303, 6.27758126451646121187334014346, 6.29147724460214324233297888925, 6.31936931835528091249048309640, 6.83078533784247317800508408900
