| L(s) = 1 | − 3·4-s − 3·9-s + 6·11-s + 6·16-s + 6·19-s − 24·29-s + 9·36-s + 18·41-s − 18·44-s − 24·59-s − 12·61-s − 10·64-s + 36·71-s − 18·76-s − 48·79-s + 6·81-s − 12·89-s − 18·99-s + 48·101-s − 36·109-s + 72·116-s + 15·121-s − 20·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 3/2·4-s − 9-s + 1.80·11-s + 3/2·16-s + 1.37·19-s − 4.45·29-s + 3/2·36-s + 2.81·41-s − 2.71·44-s − 3.12·59-s − 1.53·61-s − 5/4·64-s + 4.27·71-s − 2.06·76-s − 5.40·79-s + 2/3·81-s − 1.27·89-s − 1.80·99-s + 4.77·101-s − 3.44·109-s + 6.68·116-s + 1.36·121-s − 1.78·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1559527314\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1559527314\) |
| \(L(\frac{3}{2})\) |
|
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 | \( ( 1 + T^{2} )^{3} \) |
| 5 | \( 1 + 4 p T^{3} + p^{3} T^{6} \) |
| 7 | \( 1 \) |
| good | 11 | \( ( 1 - 3 T + 6 T^{2} - 17 T^{3} + 6 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 - 45 T^{2} + 1107 T^{4} - 17390 T^{6} + 1107 p^{2} T^{8} - 45 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 + 30 T^{2} + 927 T^{4} + 900 p T^{6} + 927 p^{2} T^{8} + 30 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( ( 1 - 3 T + 45 T^{2} - 90 T^{3} + 45 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 23 | \( 1 - 81 T^{2} + 2979 T^{4} - 75606 T^{6} + 2979 p^{2} T^{8} - 81 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 + 12 T + 120 T^{2} + 720 T^{3} + 120 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 + 78 T^{2} + 10 T^{3} + 78 p T^{4} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 165 T^{2} + 12387 T^{4} - 566110 T^{6} + 12387 p^{2} T^{8} - 165 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 - 9 T + 75 T^{2} - 610 T^{3} + 75 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{3} \) |
| 47 | \( 1 - 9 T^{2} + 1659 T^{4} - 160614 T^{6} + 1659 p^{2} T^{8} - 9 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 51 T^{2} + 8814 T^{4} - 284551 T^{6} + 8814 p^{2} T^{8} - 51 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 + 12 T + 150 T^{2} + 980 T^{3} + 150 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 + 6 T + 75 T^{2} + 20 T^{3} + 75 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 150 T^{2} + 16167 T^{4} - 1267540 T^{6} + 16167 p^{2} T^{8} - 150 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 - 6 T + p T^{2} )^{6} \) |
| 73 | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{3} \) |
| 79 | \( ( 1 + 24 T + 414 T^{2} + 4194 T^{3} + 414 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 300 T^{2} + 40392 T^{4} - 3707850 T^{6} + 40392 p^{2} T^{8} - 300 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 6 T + 159 T^{2} + 676 T^{3} + 159 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 240 T^{2} + 16752 T^{4} - 546370 T^{6} + 16752 p^{2} T^{8} - 240 p^{4} T^{10} + p^{6} T^{12} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.76584072817582689844934517498, −4.72639632667525787010073614629, −4.69384067947268735732493724970, −4.59518940668234035355814591122, −4.49836440968237360086329255152, −4.02997512767432160506861697154, −3.97883123421067713442361256920, −3.95994226688819393580873393130, −3.72577420226629408537112399697, −3.59072745456594282346403574202, −3.46436355281777437526993226962, −3.17373679532739755714162666024, −3.12125916206980249582683792413, −3.07252777023641013411064095268, −2.56240057973805297866905676386, −2.49401530361223631991477100296, −2.20998805239356689232886078334, −2.18176813408522913471207217272, −1.65488935182073369090914466814, −1.56445677607076047025220064614, −1.42832564387342996842220239744, −1.07991343466882245388602156908, −0.990336323856238504529014145970, −0.46000313344161025565679255687, −0.07628287618373447445220759771,
0.07628287618373447445220759771, 0.46000313344161025565679255687, 0.990336323856238504529014145970, 1.07991343466882245388602156908, 1.42832564387342996842220239744, 1.56445677607076047025220064614, 1.65488935182073369090914466814, 2.18176813408522913471207217272, 2.20998805239356689232886078334, 2.49401530361223631991477100296, 2.56240057973805297866905676386, 3.07252777023641013411064095268, 3.12125916206980249582683792413, 3.17373679532739755714162666024, 3.46436355281777437526993226962, 3.59072745456594282346403574202, 3.72577420226629408537112399697, 3.95994226688819393580873393130, 3.97883123421067713442361256920, 4.02997512767432160506861697154, 4.49836440968237360086329255152, 4.59518940668234035355814591122, 4.69384067947268735732493724970, 4.72639632667525787010073614629, 4.76584072817582689844934517498
Plot not available for L-functions of degree greater than 10.
