| L(s) = 1 | − 2-s − 3·3-s − 6·5-s + 3·6-s − 4·7-s + 8-s + 6·9-s + 6·10-s − 6·13-s + 4·14-s + 18·15-s − 16-s − 12·17-s − 6·18-s + 19-s + 12·21-s − 3·24-s + 19·25-s + 6·26-s − 9·27-s + 6·29-s − 18·30-s + 12·34-s + 24·35-s − 38-s + 18·39-s − 6·40-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s − 2.68·5-s + 1.22·6-s − 1.51·7-s + 0.353·8-s + 2·9-s + 1.89·10-s − 1.66·13-s + 1.06·14-s + 4.64·15-s − 1/4·16-s − 2.91·17-s − 1.41·18-s + 0.229·19-s + 2.61·21-s − 0.612·24-s + 19/5·25-s + 1.17·26-s − 1.73·27-s + 1.11·29-s − 3.28·30-s + 2.05·34-s + 4.05·35-s − 0.162·38-s + 2.88·39-s − 0.948·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12996 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 19 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 12 T + 65 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 6 T + 59 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 9 T + 94 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 5 T - 48 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 24 T + 271 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 139 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 15 T + 172 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
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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
−12.90827632006148094617876878745, −12.64180788202229429145452644977, −12.10867606907445232844022519677, −11.76088000692072397196604889310, −11.18412967965461406980137132853, −11.01410673724088045363400974489, −10.20701896490707379654839101008, −9.817584436886253214074820769885, −8.998245159580685822868552706014, −8.522355992659609080579137339979, −7.56110063126240219339484692946, −7.34965980248173268667283238865, −6.57667224547453187125174561201, −6.46395096170721997174046984607, −4.99762765579652302841456530831, −4.50093324347433417577487718006, −4.09377601857051945711254367337, −2.92474174787155679446208324715, 0, 0,
2.92474174787155679446208324715, 4.09377601857051945711254367337, 4.50093324347433417577487718006, 4.99762765579652302841456530831, 6.46395096170721997174046984607, 6.57667224547453187125174561201, 7.34965980248173268667283238865, 7.56110063126240219339484692946, 8.522355992659609080579137339979, 8.998245159580685822868552706014, 9.817584436886253214074820769885, 10.20701896490707379654839101008, 11.01410673724088045363400974489, 11.18412967965461406980137132853, 11.76088000692072397196604889310, 12.10867606907445232844022519677, 12.64180788202229429145452644977, 12.90827632006148094617876878745
