| L(s) = 1 | − 5·5-s + 28·7-s − 24·11-s + 70·13-s − 204·17-s + 40·19-s − 72·23-s + 306·29-s + 136·31-s − 140·35-s − 428·37-s − 150·41-s + 292·43-s − 72·47-s + 343·49-s + 828·53-s + 120·55-s − 744·59-s + 418·61-s − 350·65-s − 188·67-s − 960·71-s + 868·73-s − 672·77-s − 1.35e3·79-s − 612·83-s + 1.02e3·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.51·7-s − 0.657·11-s + 1.49·13-s − 2.91·17-s + 0.482·19-s − 0.652·23-s + 1.95·29-s + 0.787·31-s − 0.676·35-s − 1.90·37-s − 0.571·41-s + 1.03·43-s − 0.223·47-s + 49-s + 2.14·53-s + 0.294·55-s − 1.64·59-s + 0.877·61-s − 0.667·65-s − 0.342·67-s − 1.60·71-s + 1.39·73-s − 0.994·77-s − 1.92·79-s − 0.809·83-s + 1.30·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8806973499\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8806973499\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 5 p T + p^{3} T^{2} )( 1 + p T + p^{3} T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 24 T - 755 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 89 T + p^{3} T^{2} )( 1 + 19 T + p^{3} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 6 p T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 20 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 72 T - 6983 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 306 T + 69247 T^{2} - 306 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 136 T - 11295 T^{2} - 136 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 214 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 150 T - 46421 T^{2} + 150 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 292 T + 5757 T^{2} - 292 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 72 T - 98639 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 414 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 744 T + 348157 T^{2} + 744 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 418 T - 52257 T^{2} - 418 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 188 T - 265419 T^{2} + 188 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 480 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 434 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 1352 T + 1334865 T^{2} + 1352 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 612 T - 197243 T^{2} + 612 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 30 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 286 T - 830877 T^{2} - 286 p^{3} T^{3} + p^{6} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.110082105687472034628756895982, −8.547260882535689327380156975654, −8.540068900937614714903440636934, −8.025601777667353844490279152195, −7.991085724660757792613368335997, −6.98870363129822896151130558666, −6.95758157987109535433013259062, −6.55921888067703625589068338815, −5.96356101105182161632133712485, −5.44787944422769105583285265215, −5.14147078489080546128051608332, −4.48188421458624591445003424693, −4.28141120490515930946244434280, −4.01645323185992284302894424068, −3.21065100287582212912150922592, −2.54985571187617264689192164446, −2.31247405426533283606355487836, −1.37140440519123501488102033210, −1.30650910786750581001073580124, −0.20544300134166669683436031814,
0.20544300134166669683436031814, 1.30650910786750581001073580124, 1.37140440519123501488102033210, 2.31247405426533283606355487836, 2.54985571187617264689192164446, 3.21065100287582212912150922592, 4.01645323185992284302894424068, 4.28141120490515930946244434280, 4.48188421458624591445003424693, 5.14147078489080546128051608332, 5.44787944422769105583285265215, 5.96356101105182161632133712485, 6.55921888067703625589068338815, 6.95758157987109535433013259062, 6.98870363129822896151130558666, 7.991085724660757792613368335997, 8.025601777667353844490279152195, 8.540068900937614714903440636934, 8.547260882535689327380156975654, 9.110082105687472034628756895982
