| L(s) = 1 | − 2-s − 3-s − 4-s − 5-s + 6-s + 2·7-s + 8-s − 2·9-s + 10-s − 2·11-s + 12-s − 2·14-s + 15-s − 16-s + 17-s + 2·18-s + 20-s − 2·21-s + 2·22-s − 12·23-s − 24-s − 4·25-s + 2·27-s − 2·28-s − 3·29-s − 30-s − 2·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.755·7-s + 0.353·8-s − 2/3·9-s + 0.316·10-s − 0.603·11-s + 0.288·12-s − 0.534·14-s + 0.258·15-s − 1/4·16-s + 0.242·17-s + 0.471·18-s + 0.223·20-s − 0.436·21-s + 0.426·22-s − 2.50·23-s − 0.204·24-s − 4/5·25-s + 0.384·27-s − 0.377·28-s − 0.557·29-s − 0.182·30-s − 0.359·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10095 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10095 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 673 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 6 T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 - T + 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 12 T + 70 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3 T + 56 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 16 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 3 T + 42 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 18 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + T + 70 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 10 T + 150 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 7 T + 152 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 146 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−16.8661367364, −16.4773633397, −15.9118560126, −15.5386024032, −14.8360051288, −14.2816315346, −14.0302706285, −13.2950894801, −12.8463800753, −11.9604384843, −11.7842395088, −11.2544235088, −10.7194147959, −10.0256009846, −9.59082806379, −8.96757704173, −8.24678307715, −7.87600587563, −7.56747867369, −6.14223032740, −6.01073888574, −4.99657602241, −4.43566694543, −3.49479136533, −2.09176931954, 0,
2.09176931954, 3.49479136533, 4.43566694543, 4.99657602241, 6.01073888574, 6.14223032740, 7.56747867369, 7.87600587563, 8.24678307715, 8.96757704173, 9.59082806379, 10.0256009846, 10.7194147959, 11.2544235088, 11.7842395088, 11.9604384843, 12.8463800753, 13.2950894801, 14.0302706285, 14.2816315346, 14.8360051288, 15.5386024032, 15.9118560126, 16.4773633397, 16.8661367364
