| L(s) = 1 | + 5·4-s + 6·9-s − 16·11-s + 10·16-s + 6·19-s + 20·29-s + 8·31-s + 30·36-s − 4·41-s − 80·44-s + 10·49-s + 40·59-s − 4·61-s + 10·64-s − 8·71-s + 30·76-s + 13·81-s − 4·89-s − 96·99-s − 36·101-s + 20·109-s + 100·116-s + 110·121-s + 40·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 5/2·4-s + 2·9-s − 4.82·11-s + 5/2·16-s + 1.37·19-s + 3.71·29-s + 1.43·31-s + 5·36-s − 0.624·41-s − 12.0·44-s + 10/7·49-s + 5.20·59-s − 0.512·61-s + 5/4·64-s − 0.949·71-s + 3.44·76-s + 13/9·81-s − 0.423·89-s − 9.64·99-s − 3.58·101-s + 1.91·109-s + 9.28·116-s + 10·121-s + 3.59·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.950327599\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.950327599\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( ( 1 - T )^{6} \) |
| good | 2 | \( 1 - 5 T^{2} + 15 T^{4} - 35 T^{6} + 15 p^{2} T^{8} - 5 p^{4} T^{10} + p^{6} T^{12} \) |
| 3 | \( 1 - 2 p T^{2} + 23 T^{4} - 68 T^{6} + 23 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} \) |
| 7 | \( 1 - 10 T^{2} + 95 T^{4} - 780 T^{6} + 95 p^{2} T^{8} - 10 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 + 8 T + 41 T^{2} + 160 T^{3} + 41 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 - 38 T^{2} + 535 T^{4} - 5444 T^{6} + 535 p^{2} T^{8} - 38 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 - 26 T^{2} + 879 T^{4} - 13868 T^{6} + 879 p^{2} T^{8} - 26 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( 1 - 106 T^{2} + 5183 T^{4} - 150348 T^{6} + 5183 p^{2} T^{8} - 106 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 - 10 T + 99 T^{2} - 540 T^{3} + 99 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 - 4 T + 45 T^{2} - 184 T^{3} + 45 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 70 T^{2} + 3655 T^{4} - 120580 T^{6} + 3655 p^{2} T^{8} - 70 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 + 2 T + 87 T^{2} + 60 T^{3} + 87 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 + 46 T^{2} + 919 T^{4} - 57692 T^{6} + 919 p^{2} T^{8} + 46 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 250 T^{2} + 27375 T^{4} - 1676140 T^{6} + 27375 p^{2} T^{8} - 250 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 214 T^{2} + 22919 T^{4} - 1516452 T^{6} + 22919 p^{2} T^{8} - 214 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 - 20 T + 289 T^{2} - 2520 T^{3} + 289 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 + 2 T + 99 T^{2} + 476 T^{3} + 99 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 246 T^{2} + 30839 T^{4} - 2511908 T^{6} + 30839 p^{2} T^{8} - 246 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 + 4 T + 133 T^{2} + 504 T^{3} + 133 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 394 T^{2} + 67519 T^{4} - 6436492 T^{6} + 67519 p^{2} T^{8} - 394 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 45 T^{2} + 160 T^{3} + 45 p T^{4} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 130 T^{2} + 20135 T^{4} - 1545660 T^{6} + 20135 p^{2} T^{8} - 130 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 2 T + 135 T^{2} - 324 T^{3} + 135 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 62 T^{2} + 12895 T^{4} - 104756 T^{6} + 12895 p^{2} T^{8} - 62 p^{4} T^{10} + p^{6} T^{12} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.91032526669856359768868508896, −5.88588574564208173291060708990, −5.45141472762195219990169313067, −5.43247933869765155502925692981, −5.33877303064088435315633766897, −5.16435559488689342539680777158, −4.91071574355800502009757620631, −4.70864639395934645839245544882, −4.45394485049696642444098347691, −4.39308131613947595203654810397, −4.27572690648571192689471363043, −3.97165143462199310648785119715, −3.53717722217027165533086682704, −3.30449075657326734063347055999, −3.10221209064181506472385403439, −2.90341409623607445074803384871, −2.79554454674012482179998116117, −2.54237713377928908844146120341, −2.49977397895818048729073091438, −2.17654260545884217218747931326, −2.03121275277330042920183959218, −1.79980934376102832186888373327, −1.13577466378416316519171220224, −0.996035649149581764977218683675, −0.60232241636477317237430566922,
0.60232241636477317237430566922, 0.996035649149581764977218683675, 1.13577466378416316519171220224, 1.79980934376102832186888373327, 2.03121275277330042920183959218, 2.17654260545884217218747931326, 2.49977397895818048729073091438, 2.54237713377928908844146120341, 2.79554454674012482179998116117, 2.90341409623607445074803384871, 3.10221209064181506472385403439, 3.30449075657326734063347055999, 3.53717722217027165533086682704, 3.97165143462199310648785119715, 4.27572690648571192689471363043, 4.39308131613947595203654810397, 4.45394485049696642444098347691, 4.70864639395934645839245544882, 4.91071574355800502009757620631, 5.16435559488689342539680777158, 5.33877303064088435315633766897, 5.43247933869765155502925692981, 5.45141472762195219990169313067, 5.88588574564208173291060708990, 5.91032526669856359768868508896
Plot not available for L-functions of degree greater than 10.
