| L(s) = 1 | − 2-s − 3·3-s − 2·4-s − 2·5-s + 3·6-s − 7-s + 3·8-s + 3·9-s + 2·10-s + 11-s + 6·12-s − 2·13-s + 14-s + 6·15-s + 16-s + 17-s − 3·18-s − 2·19-s + 4·20-s + 3·21-s − 22-s − 3·23-s − 9·24-s + 25-s + 2·26-s + 2·28-s − 6·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s − 4-s − 0.894·5-s + 1.22·6-s − 0.377·7-s + 1.06·8-s + 9-s + 0.632·10-s + 0.301·11-s + 1.73·12-s − 0.554·13-s + 0.267·14-s + 1.54·15-s + 1/4·16-s + 0.242·17-s − 0.707·18-s − 0.458·19-s + 0.894·20-s + 0.654·21-s − 0.213·22-s − 0.625·23-s − 1.83·24-s + 1/5·25-s + 0.392·26-s + 0.377·28-s − 1.09·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1519 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1519 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 11 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $D_{4}$ | \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - T - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + p T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 78 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 7 T + 42 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 3 T + 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 T + 79 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 11 T + 86 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 3 T - 62 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 11 T + 102 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + T - 76 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + T + 80 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2 T - 82 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 7 T + 64 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - T + p T^{2} ) \) |
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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
−19.2838543194, −18.9180506906, −18.2659420984, −17.9603476504, −17.4457044075, −16.7899227307, −16.6376413040, −16.1215262817, −15.2085462244, −14.6748973322, −13.9560995505, −13.0471888346, −12.6916205512, −11.7999974519, −11.6323179436, −10.8078414143, −10.2380344311, −9.47669476380, −8.90147213363, −8.01984494357, −7.33784229568, −6.32627353655, −5.54341957287, −4.71649673170, −3.83210316757, 0,
3.83210316757, 4.71649673170, 5.54341957287, 6.32627353655, 7.33784229568, 8.01984494357, 8.90147213363, 9.47669476380, 10.2380344311, 10.8078414143, 11.6323179436, 11.7999974519, 12.6916205512, 13.0471888346, 13.9560995505, 14.6748973322, 15.2085462244, 16.1215262817, 16.6376413040, 16.7899227307, 17.4457044075, 17.9603476504, 18.2659420984, 18.9180506906, 19.2838543194
