| L(s) = 1 | − 4·11-s − 20·19-s + 44·59-s − 12·61-s − 64·71-s + 12·79-s + 81-s − 36·89-s + 12·101-s − 48·109-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 1.20·11-s − 4.58·19-s + 5.72·59-s − 1.53·61-s − 7.59·71-s + 1.35·79-s + 1/9·81-s − 3.81·89-s + 1.19·101-s − 4.59·109-s + 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9158711673\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9158711673\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T^{4} + T^{8} \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 23 T^{4} + p^{4} T^{8} \) |
| good | 11 | \( ( 1 + 2 T - 16 T^{2} - 4 T^{3} + 235 T^{4} - 4 p T^{5} - 16 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + 287 T^{4} + p^{4} T^{8} )^{2} \) |
| 17 | \( 1 + 60 T^{2} + 2042 T^{4} + 50520 T^{6} + 972243 T^{8} + 50520 p^{2} T^{10} + 2042 p^{4} T^{12} + 60 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( ( 1 + 10 T + 49 T^{2} + 130 T^{3} + 340 T^{4} + 130 p T^{5} + 49 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( 1 - 96 T^{2} + 4574 T^{4} - 144192 T^{6} + 3601251 T^{8} - 144192 p^{2} T^{10} + 4574 p^{4} T^{12} - 96 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 + 58 T^{2} + 2403 T^{4} + 58 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( 1 + 216 T^{2} + 22057 T^{4} + 1405080 T^{6} + 61731552 T^{8} + 1405080 p^{2} T^{10} + 22057 p^{4} T^{12} + 216 p^{6} T^{14} + p^{8} T^{16} \) |
| 41 | \( ( 1 - 108 T^{2} + 6170 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 + 868 T^{4} - 3308250 T^{8} + 868 p^{4} T^{12} + p^{8} T^{16} \) |
| 47 | \( 1 - 180 T^{2} + 16154 T^{4} - 963720 T^{6} + 47642835 T^{8} - 963720 p^{2} T^{10} + 16154 p^{4} T^{12} - 180 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( 1 + 84 T^{2} + 5642 T^{4} + 276360 T^{6} + 9540387 T^{8} + 276360 p^{2} T^{10} + 5642 p^{4} T^{12} + 84 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( ( 1 - 22 T + 248 T^{2} - 44 p T^{3} + 401 p T^{4} - 44 p^{2} T^{5} + 248 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 6 T + 73 T^{2} + 6 p T^{3} + 12 p T^{4} + 6 p^{2} T^{5} + 73 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 + 1753 T^{4} - 17078112 T^{8} + 1753 p^{4} T^{12} + p^{8} T^{16} \) |
| 71 | \( ( 1 + 8 T + p T^{2} )^{8} \) |
| 73 | \( 1 - 360 T^{2} + 64489 T^{4} - 7664040 T^{6} + 655036080 T^{8} - 7664040 p^{2} T^{10} + 64489 p^{4} T^{12} - 360 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( ( 1 - 6 T + 157 T^{2} - 870 T^{3} + 15732 T^{4} - 870 p T^{5} + 157 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 + 20092 T^{4} + 185593446 T^{8} + 20092 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( ( 1 + 18 T + 68 T^{2} + 1404 T^{3} + 28779 T^{4} + 1404 p T^{5} + 68 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 97 | \( 1 + 2446 T^{4} + 12136179 T^{8} + 2446 p^{4} T^{12} + p^{8} T^{16} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.79323042015675158660933695611, −3.75663927298155704793146193764, −3.62685592714200571745455462834, −3.36596638557723930321306645490, −3.35430790404795764557017811101, −3.14508449324439990405679209345, −2.98190122573862929690182303810, −2.91667464921569675735122889909, −2.79031220698645657984660821903, −2.72230520969319392359298530371, −2.71559691468530690649417864914, −2.26288861632155263591488806445, −2.16375054847577028412571699548, −2.13399422519919696383978083956, −2.09359700880643127037920452379, −1.89362258429622780451814927028, −1.88596477509227833786697171264, −1.65304847822068120487576734936, −1.30606674164815326137401842046, −1.20407685323599104535801054148, −1.16403417147811443292485614400, −0.72970171589273981011413560994, −0.53486422780931755219098766647, −0.28269942189794972665676783894, −0.15801739954213586683814156182,
0.15801739954213586683814156182, 0.28269942189794972665676783894, 0.53486422780931755219098766647, 0.72970171589273981011413560994, 1.16403417147811443292485614400, 1.20407685323599104535801054148, 1.30606674164815326137401842046, 1.65304847822068120487576734936, 1.88596477509227833786697171264, 1.89362258429622780451814927028, 2.09359700880643127037920452379, 2.13399422519919696383978083956, 2.16375054847577028412571699548, 2.26288861632155263591488806445, 2.71559691468530690649417864914, 2.72230520969319392359298530371, 2.79031220698645657984660821903, 2.91667464921569675735122889909, 2.98190122573862929690182303810, 3.14508449324439990405679209345, 3.35430790404795764557017811101, 3.36596638557723930321306645490, 3.62685592714200571745455462834, 3.75663927298155704793146193764, 3.79323042015675158660933695611
Plot not available for L-functions of degree greater than 10.
