| L(s) = 1 | + 64·2-s − 426·3-s + 3.07e3·4-s + 2.29e3·5-s − 2.72e4·6-s − 8.63e4·7-s + 1.31e5·8-s − 2.75e4·9-s + 1.46e5·10-s − 3.22e5·11-s − 1.30e6·12-s − 2.10e6·13-s − 5.52e6·14-s − 9.75e5·15-s + 5.24e6·16-s − 2.88e6·17-s − 1.76e6·18-s − 1.95e7·19-s + 7.03e6·20-s + 3.67e7·21-s − 2.06e7·22-s + 1.26e7·23-s − 5.58e7·24-s − 4.07e7·25-s − 1.34e8·26-s + 2.53e7·27-s − 2.65e8·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.01·3-s + 3/2·4-s + 0.327·5-s − 1.43·6-s − 1.94·7-s + 1.41·8-s − 0.155·9-s + 0.463·10-s − 0.603·11-s − 1.51·12-s − 1.56·13-s − 2.74·14-s − 0.331·15-s + 5/4·16-s − 0.492·17-s − 0.219·18-s − 1.81·19-s + 0.491·20-s + 1.96·21-s − 0.852·22-s + 0.410·23-s − 1.43·24-s − 0.834·25-s − 2.21·26-s + 0.339·27-s − 2.91·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s+11/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{5} T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + p^{5} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 142 p T + 7741 p^{3} T^{2} + 142 p^{12} T^{3} + p^{22} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 458 p T + 1840291 p^{2} T^{2} - 458 p^{12} T^{3} + p^{22} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 12332 p T + 5240877330 T^{2} + 12332 p^{12} T^{3} + p^{22} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2100184 T + 3031952118074 T^{2} + 2100184 p^{11} T^{3} + p^{22} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2882276 T + 62188541127526 T^{2} + 2882276 p^{11} T^{3} + p^{22} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 19571712 T + 327883644494230 T^{2} + 19571712 p^{11} T^{3} + p^{22} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 12680534 T + 1705711199562247 T^{2} - 12680534 p^{11} T^{3} + p^{22} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 45662496 T + 21179339599363786 T^{2} - 45662496 p^{11} T^{3} + p^{22} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 506504170 T + 114187742824378487 T^{2} + 506504170 p^{11} T^{3} + p^{22} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 402672518 T + 284017789497870851 T^{2} - 402672518 p^{11} T^{3} + p^{22} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 608864016 T + 427507291180888690 T^{2} - 608864016 p^{11} T^{3} + p^{22} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 1100094564 T + 776360461440501034 T^{2} - 1100094564 p^{11} T^{3} + p^{22} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 1012342272 T + 709446719476370206 T^{2} - 1012342272 p^{11} T^{3} + p^{22} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 68189276 T + 5575175257378167838 T^{2} + 68189276 p^{11} T^{3} + p^{22} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6791617518 T + 61459236505844701015 T^{2} + 6791617518 p^{11} T^{3} + p^{22} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 5704046520 T + 26335031748690586522 T^{2} - 5704046520 p^{11} T^{3} + p^{22} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 36514311702 T + \)\(57\!\cdots\!63\)\( T^{2} + 36514311702 p^{11} T^{3} + p^{22} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 20672196594 T + \)\(28\!\cdots\!35\)\( T^{2} - 20672196594 p^{11} T^{3} + p^{22} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 3082870856 T + \)\(57\!\cdots\!74\)\( T^{2} - 3082870856 p^{11} T^{3} + p^{22} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 28681382180 T + \)\(17\!\cdots\!22\)\( T^{2} + 28681382180 p^{11} T^{3} + p^{22} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 29532640772 T + \)\(26\!\cdots\!34\)\( T^{2} - 29532640772 p^{11} T^{3} + p^{22} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 85063742462 T + \)\(73\!\cdots\!63\)\( T^{2} - 85063742462 p^{11} T^{3} + p^{22} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 180832449678 T + \)\(22\!\cdots\!27\)\( T^{2} + 180832449678 p^{11} T^{3} + p^{22} T^{4} \) |
| 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
−14.77683333704028662987915760567, −14.66444452531371095447823722976, −13.37607664322959305095806805924, −13.12390774894013837619612237871, −12.38525435850532855675697977830, −12.28398009653442541234962044717, −10.92589417546072243124034994866, −10.85021730670769970003837818039, −9.782492652121297779541205584305, −9.148927155801302197190547315367, −7.54089304015631796414923950284, −6.87718042664531344447379400613, −5.97125955092506176537552499921, −5.82656584338831052157594166038, −4.77778491198025297446718313720, −3.86837571187162189783425874570, −2.79383779579719421555766715315, −2.16848816833170491430239796442, 0, 0,
2.16848816833170491430239796442, 2.79383779579719421555766715315, 3.86837571187162189783425874570, 4.77778491198025297446718313720, 5.82656584338831052157594166038, 5.97125955092506176537552499921, 6.87718042664531344447379400613, 7.54089304015631796414923950284, 9.148927155801302197190547315367, 9.782492652121297779541205584305, 10.85021730670769970003837818039, 10.92589417546072243124034994866, 12.28398009653442541234962044717, 12.38525435850532855675697977830, 13.12390774894013837619612237871, 13.37607664322959305095806805924, 14.66444452531371095447823722976, 14.77683333704028662987915760567
