| L(s) = 1 | − 8·5-s + 24·7-s − 2·11-s − 32·13-s − 56·17-s − 184·19-s + 92·23-s + 95·25-s + 336·29-s + 376·31-s − 192·35-s − 348·37-s − 312·41-s − 80·43-s − 228·47-s + 43·49-s − 152·53-s + 16·55-s − 680·59-s + 112·61-s + 256·65-s − 352·67-s + 1.81e3·71-s − 574·73-s − 48·77-s − 1.36e3·79-s + 782·83-s + ⋯ |
| L(s) = 1 | − 0.715·5-s + 1.29·7-s − 0.0548·11-s − 0.682·13-s − 0.798·17-s − 2.22·19-s + 0.834·23-s + 0.759·25-s + 2.15·29-s + 2.17·31-s − 0.927·35-s − 1.54·37-s − 1.18·41-s − 0.283·43-s − 0.707·47-s + 0.125·49-s − 0.393·53-s + 0.0392·55-s − 1.50·59-s + 0.235·61-s + 0.488·65-s − 0.641·67-s + 3.03·71-s − 0.920·73-s − 0.0710·77-s − 1.93·79-s + 1.03·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2985984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2985984 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 + 8 T - 31 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 24 T + 533 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T + p^{3} T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 32 T - 102 T^{2} + 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 56 T + 5858 T^{2} + 56 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 184 T + 20994 T^{2} + 184 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 p T + 21698 T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 336 T + 75814 T^{2} - 336 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 376 T + 87501 T^{2} - 376 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 348 T + 126830 T^{2} + 348 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 312 T + 132478 T^{2} + 312 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 80 T + 102402 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 228 T + 201634 T^{2} + 228 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 152 T + 253337 T^{2} + 152 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 680 T + 355286 T^{2} + 680 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 112 T + 152970 T^{2} - 112 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 352 T + 289170 T^{2} + 352 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1816 T + 1497518 T^{2} - 1816 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 287 T + p^{3} T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 1360 T + 1144350 T^{2} + 1360 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 782 T + 992327 T^{2} - 782 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 240 T + 1328110 T^{2} - 240 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 338 T + 1834899 T^{2} + 338 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.653475324924248584154504768447, −8.423122263030496840063932474039, −8.038737312925722426021986115089, −7.77600957090817442616037463632, −7.09387993654683066226247218624, −6.75154203655788331052208758541, −6.32132373370388119982003324448, −6.31037422770461497767873537549, −5.10723134332312095090963370048, −4.91837041179622731120639773536, −4.77537879151940010646297112351, −4.34086824143187473133003442522, −3.79045998428755837305086623597, −3.18771094713374450342028958896, −2.45567561717063502286341886848, −2.41744807492092059218808897967, −1.42861018007671304417395400749, −1.17138704179691235271834318609, 0, 0,
1.17138704179691235271834318609, 1.42861018007671304417395400749, 2.41744807492092059218808897967, 2.45567561717063502286341886848, 3.18771094713374450342028958896, 3.79045998428755837305086623597, 4.34086824143187473133003442522, 4.77537879151940010646297112351, 4.91837041179622731120639773536, 5.10723134332312095090963370048, 6.31037422770461497767873537549, 6.32132373370388119982003324448, 6.75154203655788331052208758541, 7.09387993654683066226247218624, 7.77600957090817442616037463632, 8.038737312925722426021986115089, 8.423122263030496840063932474039, 8.653475324924248584154504768447
