| L(s) = 1 | + 2·2-s − 5·5-s + 13·7-s − 8·8-s − 10·10-s − 30·11-s + 61·13-s + 26·14-s − 16·16-s − 24·17-s − 98·19-s − 60·22-s + 18·23-s + 122·26-s − 186·29-s + 160·31-s − 48·34-s − 65·35-s − 182·37-s − 196·38-s + 40·40-s + 378·41-s + 268·43-s + 36·46-s + 144·47-s + 343·49-s − 1.14e3·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.447·5-s + 0.701·7-s − 0.353·8-s − 0.316·10-s − 0.822·11-s + 1.30·13-s + 0.496·14-s − 1/4·16-s − 0.342·17-s − 1.18·19-s − 0.581·22-s + 0.163·23-s + 0.920·26-s − 1.19·29-s + 0.926·31-s − 0.242·34-s − 0.313·35-s − 0.808·37-s − 0.836·38-s + 0.158·40-s + 1.43·41-s + 0.950·43-s + 0.115·46-s + 0.446·47-s + 49-s − 2.95·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.968228408\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.968228408\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 13 T - 174 T^{2} - 13 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 30 T - 431 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 61 T + 1524 T^{2} - 61 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 12 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 49 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 18 T - 11843 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 186 T + 10207 T^{2} + 186 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 160 T - 4191 T^{2} - 160 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 91 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 378 T + 73963 T^{2} - 378 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 268 T - 7683 T^{2} - 268 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 144 T - 83087 T^{2} - 144 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 570 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 204 T - 163763 T^{2} - 204 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 877 T + 542148 T^{2} - 877 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 187 T - 265794 T^{2} - 187 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 606 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 431 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 1151 T + 831762 T^{2} + 1151 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 102 T - 561383 T^{2} - 102 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 984 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 265 T - 842448 T^{2} - 265 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.01332270960099688817827819702, −9.753140660791585664913237596517, −9.146569516561081907129803724991, −8.594750303052224534446509784490, −8.348615628113532852587654507799, −8.145708006717283551159484393413, −7.28111580912440758886931503730, −7.27112202969919724163430326340, −6.42304786264502955777204880686, −6.03326426346443778690420261439, −5.68318051106950629583654788270, −5.12744425662125923459648681894, −4.44908330928257312304200325030, −4.42038654623191290303836902343, −3.59505545568236913066433889267, −3.39088723236665032727912807706, −2.37287680295988863705438851330, −2.13153649240579534833368947121, −1.14577623768893148799842494685, −0.44385291053414054271330296253,
0.44385291053414054271330296253, 1.14577623768893148799842494685, 2.13153649240579534833368947121, 2.37287680295988863705438851330, 3.39088723236665032727912807706, 3.59505545568236913066433889267, 4.42038654623191290303836902343, 4.44908330928257312304200325030, 5.12744425662125923459648681894, 5.68318051106950629583654788270, 6.03326426346443778690420261439, 6.42304786264502955777204880686, 7.27112202969919724163430326340, 7.28111580912440758886931503730, 8.145708006717283551159484393413, 8.348615628113532852587654507799, 8.594750303052224534446509784490, 9.146569516561081907129803724991, 9.753140660791585664913237596517, 10.01332270960099688817827819702
