| L(s) = 1 | + 3.00e3·5-s − 3.93e4·9-s + 8.64e5·13-s − 5.51e4·17-s − 3.05e7·25-s − 8.07e7·29-s − 2.12e8·37-s − 5.44e8·41-s − 1.18e8·45-s − 3.57e8·49-s − 8.88e8·53-s − 2.08e8·61-s + 2.59e9·65-s + 4.88e9·73-s + 1.16e9·81-s − 1.65e8·85-s + 2.15e10·89-s + 1.49e10·97-s + 1.21e10·101-s − 6.99e10·109-s + 5.71e10·113-s − 3.40e10·117-s + 1.16e10·121-s − 1.24e11·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 0.959·5-s − 2/3·9-s + 2.32·13-s − 0.0388·17-s − 3.12·25-s − 3.93·29-s − 3.06·37-s − 4.69·41-s − 0.639·45-s − 1.26·49-s − 2.12·53-s − 0.246·61-s + 2.23·65-s + 2.35·73-s + 1/3·81-s − 0.0372·85-s + 3.85·89-s + 1.74·97-s + 1.15·101-s − 4.54·109-s + 3.10·113-s − 1.55·117-s + 0.448·121-s − 4.09·125-s − 3.78·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+5)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(1.420712270\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.420712270\) |
| \(L(6)\) |
|
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 | $C_2$ | \( ( 1 + p^{9} T^{2} )^{2} \) |
| good | 5 | $D_{4}$ | \( ( 1 - 12 p^{3} T + 3725566 p T^{2} - 12 p^{13} T^{3} + p^{20} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 + 357640988 T^{2} + 3876072374580582 p^{2} T^{4} + 357640988 p^{20} T^{6} + p^{40} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 11621066500 T^{2} - \)\(57\!\cdots\!18\)\( T^{4} - 11621066500 p^{20} T^{6} + p^{40} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 33236 p T + 195679411734 T^{2} - 33236 p^{11} T^{3} + p^{20} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 + 27564 T - 539669051098 T^{2} + 27564 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 15556349136580 T^{2} + \)\(11\!\cdots\!22\)\( T^{4} - 15556349136580 p^{20} T^{6} + p^{40} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 132444369581380 T^{2} + \)\(77\!\cdots\!82\)\( T^{4} - 132444369581380 p^{20} T^{6} + p^{40} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 40395156 T + 1171301940969206 T^{2} + 40395156 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 3045470037697060 T^{2} + \)\(36\!\cdots\!22\)\( T^{4} - 3045470037697060 p^{20} T^{6} + p^{40} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 106255820 T + 11985453761613078 T^{2} + 106255820 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 272226636 T + 39503940013305926 T^{2} + 272226636 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 26934619838954500 T^{2} + \)\(74\!\cdots\!82\)\( T^{4} - 26934619838954500 p^{20} T^{6} + p^{40} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 77230393510956292 T^{2} + \)\(63\!\cdots\!98\)\( T^{4} - 77230393510956292 p^{20} T^{6} + p^{40} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 444402036 T + 202557268735976342 T^{2} + 444402036 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 1112740587853424260 T^{2} + \)\(82\!\cdots\!82\)\( T^{4} - 1112740587853424260 p^{20} T^{6} + p^{40} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 104167628 T + 1117708367535171318 T^{2} + 104167628 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 6706192991646129220 T^{2} + \)\(17\!\cdots\!22\)\( T^{4} - 6706192991646129220 p^{20} T^{6} + p^{40} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 1787686856744485828 T^{2} + \)\(15\!\cdots\!18\)\( T^{4} - 1787686856744485828 p^{20} T^{6} + p^{40} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 2440834724 T + 7171167497674590822 T^{2} - 2440834724 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 31243817648500267684 T^{2} + \)\(42\!\cdots\!66\)\( T^{4} - 31243817648500267684 p^{20} T^{6} + p^{40} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 35005678821449308612 T^{2} + \)\(78\!\cdots\!18\)\( T^{4} - 35005678821449308612 p^{20} T^{6} + p^{40} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 10770191268 T + 83664060888855623078 T^{2} - 10770191268 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 7482884228 T + 77690767738995235974 T^{2} - 7482884228 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.359360866834870867073179132983, −9.292259545149114541870937937792, −8.786825761509876715135981880396, −8.490174920622977460024207255727, −8.109890155848125663795214427292, −7.75561678814156486087073831588, −7.72139907455090462592022869964, −6.91924324442602423488893329484, −6.58196539170073039710313682481, −6.42258712184795806850901461522, −6.00749367857920925770467056062, −5.61173046505282851671702380050, −5.32091912350776865152470371279, −5.26403768043087518146691135722, −4.58809332837017225896294151566, −3.77789581347441083510251539661, −3.60728090836348501151622724560, −3.42638991453119779974119538926, −3.27762741534758948605038293354, −1.98419302099995261396626592497, −1.87766640719667360215074073839, −1.75822795208360437399139355236, −1.59340526080658926942323332029, −0.46710347405720724447669095769, −0.23173603768634559891881807168,
0.23173603768634559891881807168, 0.46710347405720724447669095769, 1.59340526080658926942323332029, 1.75822795208360437399139355236, 1.87766640719667360215074073839, 1.98419302099995261396626592497, 3.27762741534758948605038293354, 3.42638991453119779974119538926, 3.60728090836348501151622724560, 3.77789581347441083510251539661, 4.58809332837017225896294151566, 5.26403768043087518146691135722, 5.32091912350776865152470371279, 5.61173046505282851671702380050, 6.00749367857920925770467056062, 6.42258712184795806850901461522, 6.58196539170073039710313682481, 6.91924324442602423488893329484, 7.72139907455090462592022869964, 7.75561678814156486087073831588, 8.109890155848125663795214427292, 8.490174920622977460024207255727, 8.786825761509876715135981880396, 9.292259545149114541870937937792, 9.359360866834870867073179132983
