| L(s) = 1 | − 3·4-s + 2·5-s + 4·7-s + 6·16-s − 12·17-s + 16·19-s − 6·20-s − 28·23-s + 25-s − 12·28-s + 8·35-s − 12·37-s − 14·49-s − 10·64-s + 36·68-s − 16·73-s − 48·76-s + 12·80-s − 24·85-s + 28·89-s + 84·92-s + 32·95-s + 8·97-s − 3·100-s + 4·101-s − 24·107-s + 24·112-s + ⋯ |
| L(s) = 1 | − 3/2·4-s + 0.894·5-s + 1.51·7-s + 3/2·16-s − 2.91·17-s + 3.67·19-s − 1.34·20-s − 5.83·23-s + 1/5·25-s − 2.26·28-s + 1.35·35-s − 1.97·37-s − 2·49-s − 5/4·64-s + 4.36·68-s − 1.87·73-s − 5.50·76-s + 1.34·80-s − 2.60·85-s + 2.96·89-s + 8.75·92-s + 3.28·95-s + 0.812·97-s − 0.299·100-s + 0.398·101-s − 2.32·107-s + 2.26·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 5^{6} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 5^{6} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.421340613\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.421340613\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{3} \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 2 T + 3 T^{2} - 12 T^{3} + 3 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | \( 1 + 12 T + 83 T^{2} + 392 T^{3} + 83 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| good | 7 | \( ( 1 - 2 T + 13 T^{2} - 32 T^{3} + 13 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 11 | \( 1 - 2 p T^{2} + 375 T^{4} - 3988 T^{6} + 375 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} \) |
| 13 | \( 1 - 46 T^{2} + 1063 T^{4} - 16468 T^{6} + 1063 p^{2} T^{8} - 46 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( ( 1 - 8 T + 41 T^{2} - 144 T^{3} + 41 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 23 | \( ( 1 + 14 T + 81 T^{2} + 348 T^{3} + 81 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 29 | \( 1 - 142 T^{2} + 9095 T^{4} - 337188 T^{6} + 9095 p^{2} T^{8} - 142 p^{4} T^{10} + p^{6} T^{12} \) |
| 31 | \( 1 - 110 T^{2} + 5935 T^{4} - 214532 T^{6} + 5935 p^{2} T^{8} - 110 p^{4} T^{10} + p^{6} T^{12} \) |
| 37 | \( ( 1 + 6 T + 11 T^{2} - 188 T^{3} + 11 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 41 | \( 1 - 58 T^{2} + 2607 T^{4} - 146620 T^{6} + 2607 p^{2} T^{8} - 58 p^{4} T^{10} + p^{6} T^{12} \) |
| 43 | \( 1 - 70 T^{2} + 5095 T^{4} - 187780 T^{6} + 5095 p^{2} T^{8} - 70 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 202 T^{2} + 19503 T^{4} - 1148236 T^{6} + 19503 p^{2} T^{8} - 202 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 146 T^{2} + 13143 T^{4} - 859868 T^{6} + 13143 p^{2} T^{8} - 146 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 + 129 T^{2} - 20 T^{3} + 129 p T^{4} + p^{3} T^{6} )^{2} \) |
| 61 | \( 1 - 34 T^{2} + 7751 T^{4} - 300156 T^{6} + 7751 p^{2} T^{8} - 34 p^{4} T^{10} + p^{6} T^{12} \) |
| 67 | \( 1 - 182 T^{2} + 11095 T^{4} - 445796 T^{6} + 11095 p^{2} T^{8} - 182 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( 1 - 330 T^{2} + 50655 T^{4} - 4581372 T^{6} + 50655 p^{2} T^{8} - 330 p^{4} T^{10} + p^{6} T^{12} \) |
| 73 | \( ( 1 + 8 T + 231 T^{2} + 1172 T^{3} + 231 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 79 | \( 1 - 334 T^{2} + 51983 T^{4} - 5030340 T^{6} + 51983 p^{2} T^{8} - 334 p^{4} T^{10} + p^{6} T^{12} \) |
| 83 | \( 1 - 418 T^{2} + 77543 T^{4} - 8255484 T^{6} + 77543 p^{2} T^{8} - 418 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 - 14 T - 41 T^{2} + 1788 T^{3} - 41 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( ( 1 - 4 T + 111 T^{2} + 332 T^{3} + 111 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
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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
−4.96701288077960066902059188729, −4.79682685327190446326028091215, −4.47485260484194702314250436822, −4.43875563793580030934173836412, −4.39677692374045470043559350079, −4.27550628008566175926306229595, −4.25968219124088861732221937862, −4.00648175477281282696835611420, −3.47516004205006276952874183493, −3.44877383730942171473055497958, −3.43484362780310439635064240803, −3.38147743403927319437927062050, −3.16168106895499776263045187266, −2.82965221827142935686148393413, −2.54034493172479227377577175730, −2.19411584123825367410398613907, −2.12159288021648228023315214504, −2.06031594077829389680407827721, −1.83132940636876640589722160001, −1.67368777256257393451968391426, −1.54820063869341658619607028646, −1.19227568453891525820563101009, −0.894728640291971912543327326392, −0.30765913381509623256402599604, −0.30422633135659247154684131854,
0.30422633135659247154684131854, 0.30765913381509623256402599604, 0.894728640291971912543327326392, 1.19227568453891525820563101009, 1.54820063869341658619607028646, 1.67368777256257393451968391426, 1.83132940636876640589722160001, 2.06031594077829389680407827721, 2.12159288021648228023315214504, 2.19411584123825367410398613907, 2.54034493172479227377577175730, 2.82965221827142935686148393413, 3.16168106895499776263045187266, 3.38147743403927319437927062050, 3.43484362780310439635064240803, 3.44877383730942171473055497958, 3.47516004205006276952874183493, 4.00648175477281282696835611420, 4.25968219124088861732221937862, 4.27550628008566175926306229595, 4.39677692374045470043559350079, 4.43875563793580030934173836412, 4.47485260484194702314250436822, 4.79682685327190446326028091215, 4.96701288077960066902059188729
Plot not available for L-functions of degree greater than 10.
