| L(s) = 1 | − 2-s + 3-s − 5-s − 6-s + 8-s + 10-s − 4·13-s − 15-s − 16-s − 6·17-s − 4·19-s + 24-s + 4·26-s − 27-s − 12·29-s + 30-s − 4·31-s + 6·34-s − 2·37-s + 4·38-s − 4·39-s − 40-s − 12·41-s + 16·43-s − 12·47-s − 48-s − 6·51-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 0.447·5-s − 0.408·6-s + 0.353·8-s + 0.316·10-s − 1.10·13-s − 0.258·15-s − 1/4·16-s − 1.45·17-s − 0.917·19-s + 0.204·24-s + 0.784·26-s − 0.192·27-s − 2.22·29-s + 0.182·30-s − 0.718·31-s + 1.02·34-s − 0.328·37-s + 0.648·38-s − 0.640·39-s − 0.158·40-s − 1.87·41-s + 2.43·43-s − 1.75·47-s − 0.144·48-s − 0.840·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 16 T + 177 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−9.477079589587286030686159063108, −8.865843105224455683594419305185, −8.394091704198469552202624977872, −8.363821012557882856767768964170, −7.69670166705165295775437114777, −7.44437205670186330518797592721, −6.97655545184984978344651796544, −6.72460477317465383736458038201, −6.07886223040642050816671314100, −5.58765684193323862728272708152, −5.08926112348528690078091995382, −4.49257484236484526442643667728, −4.29691159603770646567052392066, −3.64149259350878361722586983890, −3.21182245778129604474497192912, −2.46257639871977153123855750288, −2.03104051863281352487169200531, −1.54779382087173936922044841951, 0, 0,
1.54779382087173936922044841951, 2.03104051863281352487169200531, 2.46257639871977153123855750288, 3.21182245778129604474497192912, 3.64149259350878361722586983890, 4.29691159603770646567052392066, 4.49257484236484526442643667728, 5.08926112348528690078091995382, 5.58765684193323862728272708152, 6.07886223040642050816671314100, 6.72460477317465383736458038201, 6.97655545184984978344651796544, 7.44437205670186330518797592721, 7.69670166705165295775437114777, 8.363821012557882856767768964170, 8.394091704198469552202624977872, 8.865843105224455683594419305185, 9.477079589587286030686159063108
