| L(s) = 1 | + 4·3-s − 8·7-s + 8·9-s − 8·17-s − 32·21-s + 12·27-s − 8·43-s + 16·49-s − 32·51-s + 8·53-s − 16·59-s − 64·63-s − 24·67-s − 16·71-s + 23·81-s + 8·103-s − 48·109-s − 40·113-s + 64·119-s − 28·121-s + 127-s − 32·129-s + 131-s + 137-s + 139-s + 64·147-s + 149-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 3.02·7-s + 8/3·9-s − 1.94·17-s − 6.98·21-s + 2.30·27-s − 1.21·43-s + 16/7·49-s − 4.48·51-s + 1.09·53-s − 2.08·59-s − 8.06·63-s − 2.93·67-s − 1.89·71-s + 23/9·81-s + 0.788·103-s − 4.59·109-s − 3.76·113-s + 5.86·119-s − 2.54·121-s + 0.0887·127-s − 2.81·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5.27·147-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8473810095\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8473810095\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $D_{4}$ | \( ( 1 + 4 T + 16 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_4$ | \( ( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 834 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 12 T^{2} - 1834 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - 78 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 4 T - 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 144 T^{2} + 9314 T^{4} - 144 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_{4}$ | \( ( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 90 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 12 T + 120 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 8 T + 150 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_4\times C_2$ | \( 1 - 28 T^{2} - 1946 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 12774 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 224 T^{2} + 25522 T^{4} - 224 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 8806 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_4\times C_2$ | \( 1 + 196 T^{2} + 23814 T^{4} + 196 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.46480980624158757859443512782, −6.32053136099811993179216953258, −6.04001256009573965532562972377, −5.91901269074076452941616561502, −5.48096212785585171023449008047, −5.44352064079565688152753454918, −5.06214755876157013257760103599, −4.65485412366594953714578842974, −4.65090159156254769784127566847, −4.31588032579121051688519370677, −4.13466273104950444861674502415, −3.93885803989304501445664910713, −3.65285221292098638775240322504, −3.44749087836661785750565132887, −3.19063561406820322768653836065, −3.04879792439763736152777259207, −2.81057384900488330645414989652, −2.71536806797123103896027974861, −2.60655695855997280987215329255, −2.14218418945871467103825991766, −1.79118224917832370805314497806, −1.48095251425234716265534826934, −1.43717836329220106788018700167, −0.49964425493072490125387421834, −0.16789749042485046854536161849,
0.16789749042485046854536161849, 0.49964425493072490125387421834, 1.43717836329220106788018700167, 1.48095251425234716265534826934, 1.79118224917832370805314497806, 2.14218418945871467103825991766, 2.60655695855997280987215329255, 2.71536806797123103896027974861, 2.81057384900488330645414989652, 3.04879792439763736152777259207, 3.19063561406820322768653836065, 3.44749087836661785750565132887, 3.65285221292098638775240322504, 3.93885803989304501445664910713, 4.13466273104950444861674502415, 4.31588032579121051688519370677, 4.65090159156254769784127566847, 4.65485412366594953714578842974, 5.06214755876157013257760103599, 5.44352064079565688152753454918, 5.48096212785585171023449008047, 5.91901269074076452941616561502, 6.04001256009573965532562972377, 6.32053136099811993179216953258, 6.46480980624158757859443512782
