| L(s) = 1 | − 2-s − 2·3-s − 4-s − 2·5-s + 2·6-s − 7-s + 8-s + 2·10-s − 7·11-s + 2·12-s − 6·13-s + 14-s + 4·15-s − 16-s − 2·17-s − 12·19-s + 2·20-s + 2·21-s + 7·22-s − 6·23-s − 2·24-s + 4·25-s + 6·26-s + 2·27-s + 28-s − 3·29-s − 4·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 1/2·4-s − 0.894·5-s + 0.816·6-s − 0.377·7-s + 0.353·8-s + 0.632·10-s − 2.11·11-s + 0.577·12-s − 1.66·13-s + 0.267·14-s + 1.03·15-s − 1/4·16-s − 0.485·17-s − 2.75·19-s + 0.447·20-s + 0.436·21-s + 1.49·22-s − 1.25·23-s − 0.408·24-s + 4/5·25-s + 1.17·26-s + 0.384·27-s + 0.188·28-s − 0.557·29-s − 0.730·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 771067 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 771067 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 6 T + p T^{2} ) \) |
| 191 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 16 T + p T^{2} ) \) |
| 367 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 16 T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 2 T + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 12 T + 68 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3 T + 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 5 T + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 3 T + 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 9 T + 71 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 3 T + 57 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 10 T + 82 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 70 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 13 T + 95 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + T - 44 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T + 62 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 19 T + 225 T^{2} + 19 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 7 T + 37 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 9 T + 19 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 196 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 11 T + 126 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
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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
−12.8400243468, −12.4222586744, −12.1187411158, −11.7914940594, −11.0637204250, −10.8836859764, −10.6606262329, −10.2097206660, −9.74431916678, −9.40834454492, −8.73833271152, −8.52063858546, −8.05012100198, −7.71156031232, −7.20823968045, −6.83059233116, −6.16380153658, −5.72565034210, −5.43338360077, −4.71422961843, −4.37046451156, −4.15058392953, −2.99893164443, −2.56942011351, −1.96992345800, 0, 0, 0,
1.96992345800, 2.56942011351, 2.99893164443, 4.15058392953, 4.37046451156, 4.71422961843, 5.43338360077, 5.72565034210, 6.16380153658, 6.83059233116, 7.20823968045, 7.71156031232, 8.05012100198, 8.52063858546, 8.73833271152, 9.40834454492, 9.74431916678, 10.2097206660, 10.6606262329, 10.8836859764, 11.0637204250, 11.7914940594, 12.1187411158, 12.4222586744, 12.8400243468
