| L(s) = 1 | + 4-s − 4·7-s + 4·13-s + 12·19-s − 4·28-s + 24·31-s + 12·37-s − 16·43-s + 10·49-s + 4·52-s + 3·64-s − 16·67-s − 4·73-s + 12·76-s − 16·91-s + 44·103-s + 28·109-s − 2·121-s + 24·124-s + 127-s + 131-s − 48·133-s + 137-s + 139-s + 12·148-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 1.51·7-s + 1.10·13-s + 2.75·19-s − 0.755·28-s + 4.31·31-s + 1.97·37-s − 2.43·43-s + 10/7·49-s + 0.554·52-s + 3/8·64-s − 1.95·67-s − 0.468·73-s + 1.37·76-s − 1.67·91-s + 4.33·103-s + 2.68·109-s − 0.181·121-s + 2.15·124-s + 0.0887·127-s + 0.0873·131-s − 4.16·133-s + 0.0854·137-s + 0.0848·139-s + 0.986·148-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.699949925\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.699949925\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 2 | $C_2^2 \wr C_2$ | \( 1 - T^{2} + T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 + 2 T^{2} + 127 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2 \wr C_2$ | \( 1 + 32 T^{2} + 718 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 + 50 T^{2} + 1567 T^{4} + 50 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 + 42 T^{2} + 1079 T^{4} + 42 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 + 20 T^{2} + 1606 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 8 T + 73 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 - 24 T^{2} + 3518 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 + 68 T^{2} + 4918 T^{4} + 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 + 24 T^{2} + 6062 T^{4} + 24 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 67 | $D_{4}$ | \( ( 1 + 8 T + 121 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 + 90 T^{2} + 2711 T^{4} + 90 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 2 T + 118 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 129 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 + 36 T^{2} - 2602 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 + 160 T^{2} + 12846 T^{4} + 160 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 + 78 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.64410108836379703747592469041, −6.45144494785194806789629855779, −6.34027337238007324559786609945, −6.09916501603782867757633831883, −5.93629031025666595473666199433, −5.88074145088887858900943359860, −5.28986719250112332829344298135, −5.27622717604319760196831776801, −5.10049941665954276071404646082, −4.68356009362406799924262241932, −4.36588161253616437406354148471, −4.36532863145291517270454483236, −4.15305800816045799105803153411, −3.54588930586476177264131495184, −3.49175690186987154061121356657, −3.15898221952644443036656883654, −3.03754061987304116655603212095, −2.85818869236121584095238394277, −2.75046904951728717118354115665, −2.21246428365054150094463502862, −1.84243654883089816851595569051, −1.54933564949760712281962927589, −1.01093751673236134812589070227, −0.820980777020587896750483728234, −0.58376036990521719785602097548,
0.58376036990521719785602097548, 0.820980777020587896750483728234, 1.01093751673236134812589070227, 1.54933564949760712281962927589, 1.84243654883089816851595569051, 2.21246428365054150094463502862, 2.75046904951728717118354115665, 2.85818869236121584095238394277, 3.03754061987304116655603212095, 3.15898221952644443036656883654, 3.49175690186987154061121356657, 3.54588930586476177264131495184, 4.15305800816045799105803153411, 4.36532863145291517270454483236, 4.36588161253616437406354148471, 4.68356009362406799924262241932, 5.10049941665954276071404646082, 5.27622717604319760196831776801, 5.28986719250112332829344298135, 5.88074145088887858900943359860, 5.93629031025666595473666199433, 6.09916501603782867757633831883, 6.34027337238007324559786609945, 6.45144494785194806789629855779, 6.64410108836379703747592469041
