| L(s) = 1 | + 3·4-s − 8·11-s + 5·16-s − 12·19-s + 8·29-s − 20·31-s − 12·41-s − 24·44-s + 10·49-s − 24·59-s + 4·61-s + 3·64-s − 36·76-s − 24·79-s − 28·89-s + 24·101-s + 4·109-s + 24·116-s + 26·121-s − 60·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | + 3/2·4-s − 2.41·11-s + 5/4·16-s − 2.75·19-s + 1.48·29-s − 3.59·31-s − 1.87·41-s − 3.61·44-s + 10/7·49-s − 3.12·59-s + 0.512·61-s + 3/8·64-s − 4.12·76-s − 2.70·79-s − 2.96·89-s + 2.38·101-s + 0.383·109-s + 2.22·116-s + 2.36·121-s − 5.38·124-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8555625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8555625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1521073584\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1521073584\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.842696837328822168070456740050, −8.432183105823413964062259850804, −8.260984650709569597645771600855, −7.60234051430765614882148630838, −7.53355356612732070911146823778, −7.00009446406438912127659312753, −6.85377203529618854808121645842, −6.20492204608251247810457418887, −6.00546090574766662132771827290, −5.53546138697624766168303037871, −5.22306153333215458357659172584, −4.68378257064995887687037458761, −4.29257128120266313789371610269, −3.69005198998887494817907210079, −3.16410410949470309381888198153, −2.62286568980532115017951900363, −2.50150024631790202981254708072, −1.75016680826516673529248393779, −1.68186261123444297073973559222, −0.10561412153367324341644205896,
0.10561412153367324341644205896, 1.68186261123444297073973559222, 1.75016680826516673529248393779, 2.50150024631790202981254708072, 2.62286568980532115017951900363, 3.16410410949470309381888198153, 3.69005198998887494817907210079, 4.29257128120266313789371610269, 4.68378257064995887687037458761, 5.22306153333215458357659172584, 5.53546138697624766168303037871, 6.00546090574766662132771827290, 6.20492204608251247810457418887, 6.85377203529618854808121645842, 7.00009446406438912127659312753, 7.53355356612732070911146823778, 7.60234051430765614882148630838, 8.260984650709569597645771600855, 8.432183105823413964062259850804, 8.842696837328822168070456740050
