| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.950585294\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.950585294\) |
| \(L(\frac{5}{2})\) |
|
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
−11.75652011801276212755985512762, −11.62319769527987446593571599183, −11.20935720525375387896137826962, −10.86300652756105116965916769688, −9.948406779166021267550032831902, −9.818752381762368228896260617856, −9.130683318070528314022660000477, −8.730801458151895304250466997612, −7.934063348712852077081378271170, −7.87837666580871029770478468792, −6.85990301171357867089887356814, −6.67330642341802606587757910765, −5.62660380242864224355834044808, −5.40783014961866865821024363081, −4.78086469792266745210997262291, −4.10923400974408864416891894148, −3.21768739712064095209609283749, −2.47041062609690451042820646472, −1.52798351976976298889267228951, −0.948350313388673470470569726750,
0.948350313388673470470569726750, 1.52798351976976298889267228951, 2.47041062609690451042820646472, 3.21768739712064095209609283749, 4.10923400974408864416891894148, 4.78086469792266745210997262291, 5.40783014961866865821024363081, 5.62660380242864224355834044808, 6.67330642341802606587757910765, 6.85990301171357867089887356814, 7.87837666580871029770478468792, 7.934063348712852077081378271170, 8.730801458151895304250466997612, 9.130683318070528314022660000477, 9.818752381762368228896260617856, 9.948406779166021267550032831902, 10.86300652756105116965916769688, 11.20935720525375387896137826962, 11.62319769527987446593571599183, 11.75652011801276212755985512762
