| L(s) = 1 | + 4-s − 3·9-s − 12·11-s + 4·16-s + 24·19-s − 12·31-s − 3·36-s − 12·41-s − 12·44-s − 11·49-s − 18·61-s + 11·64-s + 24·76-s − 32·79-s + 6·89-s + 36·99-s + 30·101-s − 10·109-s + 62·121-s − 12·124-s + 127-s + 131-s + 137-s + 139-s − 12·144-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 9-s − 3.61·11-s + 16-s + 5.50·19-s − 2.15·31-s − 1/2·36-s − 1.87·41-s − 1.80·44-s − 1.57·49-s − 2.30·61-s + 11/8·64-s + 2.75·76-s − 3.60·79-s + 0.635·89-s + 3.61·99-s + 2.98·101-s − 0.957·109-s + 5.63·121-s − 1.07·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 144-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9230359896\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9230359896\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| good | 2 | $C_2^3$ | \( 1 - T^{2} - 3 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 2 T^{2} - 285 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 12 T + 67 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 43 T^{2} + 1320 T^{4} - 43 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 55 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 6 T + 43 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^3$ | \( 1 + 58 T^{2} + 1995 T^{4} + 58 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 9 T + 88 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 - 35 T^{2} - 3264 T^{4} - 35 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^3$ | \( 1 - 134 T^{2} + 12627 T^{4} - 134 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 16 T + 177 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 3 T - 80 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−7.933299186870908890610461164933, −7.71225114645679270531125744911, −7.33861703284010859177609656936, −7.15624459018367723618390762043, −7.02574838814946377938812159393, −6.98688163923674590744236983753, −6.09749521862435195714747803608, −6.09405800060763669221456629470, −5.66502639374113202375159648487, −5.53359212035199242984853930380, −5.46227075140243451906717640114, −5.20055526724219058924563822022, −4.95853586100481030807384072809, −4.88554637999535604849158042519, −4.39531918756213274878655828032, −3.79571754599501872709566497319, −3.20349149156272209553026537353, −3.15791342064090526682075058326, −3.15433494293506180301455294700, −3.12520920391336131066966929690, −2.42200619049704896762995469668, −2.14061573664924536435872093846, −1.48803078637278665572929698782, −1.23537609397020519097240725519, −0.31149504492441376888974877974,
0.31149504492441376888974877974, 1.23537609397020519097240725519, 1.48803078637278665572929698782, 2.14061573664924536435872093846, 2.42200619049704896762995469668, 3.12520920391336131066966929690, 3.15433494293506180301455294700, 3.15791342064090526682075058326, 3.20349149156272209553026537353, 3.79571754599501872709566497319, 4.39531918756213274878655828032, 4.88554637999535604849158042519, 4.95853586100481030807384072809, 5.20055526724219058924563822022, 5.46227075140243451906717640114, 5.53359212035199242984853930380, 5.66502639374113202375159648487, 6.09405800060763669221456629470, 6.09749521862435195714747803608, 6.98688163923674590744236983753, 7.02574838814946377938812159393, 7.15624459018367723618390762043, 7.33861703284010859177609656936, 7.71225114645679270531125744911, 7.933299186870908890610461164933
