| L(s) = 1 | − 2·2-s − 4·5-s − 3·7-s + 4·8-s − 3·9-s + 8·10-s + 2·13-s + 6·14-s − 4·16-s + 2·17-s + 6·18-s + 5·19-s − 10·23-s + 4·25-s − 4·26-s − 2·29-s − 3·31-s − 4·34-s + 12·35-s − 37-s − 10·38-s − 16·40-s − 8·41-s + 8·43-s + 12·45-s + 20·46-s − 10·47-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.78·5-s − 1.13·7-s + 1.41·8-s − 9-s + 2.52·10-s + 0.554·13-s + 1.60·14-s − 16-s + 0.485·17-s + 1.41·18-s + 1.14·19-s − 2.08·23-s + 4/5·25-s − 0.784·26-s − 0.371·29-s − 0.538·31-s − 0.685·34-s + 2.02·35-s − 0.164·37-s − 1.62·38-s − 2.52·40-s − 1.24·41-s + 1.21·43-s + 1.78·45-s + 2.94·46-s − 1.45·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2069 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2069 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2069 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 74 T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 9 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 23 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 5 T + 13 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 + 2 T - 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 3 T - 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + T - 5 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 47 | $D_{4}$ | \( 1 + 10 T + 60 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 3 T - 10 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 3 T + 43 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 3 T + 9 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 7 T + 29 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 2 T + 124 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 14 T + 103 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 - 13 T + 106 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 3 T + 23 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 14 T + 138 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
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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
−18.9815852032, −18.6633014810, −17.9588484748, −17.7867856942, −16.8379391846, −16.4922271572, −15.9421226627, −15.7029213623, −14.8575026415, −14.0457781394, −13.7040825580, −12.8612458735, −12.0972527345, −11.7018013270, −11.1040607510, −10.2089922773, −9.69645645101, −9.15170144184, −8.32894094494, −8.02275427808, −7.49283944962, −6.38865585040, −5.37377754234, −3.98400073802, −3.42898108818, 0,
3.42898108818, 3.98400073802, 5.37377754234, 6.38865585040, 7.49283944962, 8.02275427808, 8.32894094494, 9.15170144184, 9.69645645101, 10.2089922773, 11.1040607510, 11.7018013270, 12.0972527345, 12.8612458735, 13.7040825580, 14.0457781394, 14.8575026415, 15.7029213623, 15.9421226627, 16.4922271572, 16.8379391846, 17.7867856942, 17.9588484748, 18.6633014810, 18.9815852032
