L(s) = 1 | − 2-s − 2·4-s + 7-s + 3·8-s + 11-s − 2·13-s − 14-s + 16-s − 6·17-s + 5·19-s − 22-s − 3·23-s + 2·26-s − 2·28-s + 10·29-s + 4·31-s − 2·32-s + 6·34-s + 6·37-s − 5·38-s − 4·41-s + 8·43-s − 2·44-s + 3·46-s − 21·47-s − 2·49-s + 4·52-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 4-s + 0.377·7-s + 1.06·8-s + 0.301·11-s − 0.554·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s + 1.14·19-s − 0.213·22-s − 0.625·23-s + 0.392·26-s − 0.377·28-s + 1.85·29-s + 0.718·31-s − 0.353·32-s + 1.02·34-s + 0.986·37-s − 0.811·38-s − 0.624·41-s + 1.21·43-s − 0.301·44-s + 0.442·46-s − 3.06·47-s − 2/7·49-s + 0.554·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | | \( 1 \) |
good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 - T + 21 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 5 T + 33 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3 T + 37 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 10 T + 78 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 66 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 82 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 21 T + 203 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 10 T + 63 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 11 T + 141 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 14 T + 163 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 9 T + 101 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 7 T + 147 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 113 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 5 T + 183 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 19 T + 283 T^{2} + 19 p T^{3} + p^{2} T^{4} \) |
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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
−7.997198260117927697520064820913, −7.88853698352647366217674120195, −7.35663880831290157186157207127, −6.79827079079622531146088883879, −6.71822603818540935394082891675, −6.26546482847243600976778341612, −5.75022788251178262451060567220, −5.46748248745067043543117992597, −4.78824999450365271891464673845, −4.65565157931635942122052662935, −4.41030487239414827544981280135, −4.15187376658563593192546674036, −3.36921134534008706796559270764, −2.96943269033835306802139366571, −2.65776609986496083070525828405, −1.99681700103620562341558948571, −1.26235373360684379362149506591, −1.18586945317934901529615323259, 0, 0,
1.18586945317934901529615323259, 1.26235373360684379362149506591, 1.99681700103620562341558948571, 2.65776609986496083070525828405, 2.96943269033835306802139366571, 3.36921134534008706796559270764, 4.15187376658563593192546674036, 4.41030487239414827544981280135, 4.65565157931635942122052662935, 4.78824999450365271891464673845, 5.46748248745067043543117992597, 5.75022788251178262451060567220, 6.26546482847243600976778341612, 6.71822603818540935394082891675, 6.79827079079622531146088883879, 7.35663880831290157186157207127, 7.88853698352647366217674120195, 7.997198260117927697520064820913