| L(s) = 1 | − 3·4-s − 3·9-s + 6·16-s − 8·19-s − 5·25-s + 12·29-s + 16·31-s + 9·36-s − 4·41-s + 30·49-s + 20·59-s + 20·61-s − 10·64-s − 20·71-s + 24·76-s + 8·79-s + 6·81-s + 32·89-s + 15·100-s + 4·101-s + 28·109-s − 36·116-s − 46·121-s − 48·124-s − 8·125-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 3/2·4-s − 9-s + 3/2·16-s − 1.83·19-s − 25-s + 2.22·29-s + 2.87·31-s + 3/2·36-s − 0.624·41-s + 30/7·49-s + 2.60·59-s + 2.56·61-s − 5/4·64-s − 2.37·71-s + 2.75·76-s + 0.900·79-s + 2/3·81-s + 3.39·89-s + 3/2·100-s + 0.398·101-s + 2.68·109-s − 3.34·116-s − 4.18·121-s − 4.31·124-s − 0.715·125-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.414503404\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.414503404\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} + 8 T^{3} + p^{2} T^{4} + p^{3} T^{6} \) |
| 23 | \( ( 1 + T^{2} )^{3} \) |
| good | 7 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{3} \) |
| 11 | \( ( 1 + 23 T^{2} - 8 T^{3} + 23 p T^{4} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 - 22 T^{2} + 535 T^{4} - 6772 T^{6} + 535 p^{2} T^{8} - 22 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 - 50 T^{2} + 1311 T^{4} - 25244 T^{6} + 1311 p^{2} T^{8} - 50 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( ( 1 + 4 T - 5 T^{2} - 80 T^{3} - 5 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 29 | \( ( 1 - 2 T + p T^{2} )^{6} \) |
| 31 | \( ( 1 - 8 T + 81 T^{2} - 416 T^{3} + 81 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 94 T^{2} + 143 p T^{4} - 208572 T^{6} + 143 p^{3} T^{8} - 94 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 + 2 T + 99 T^{2} + 180 T^{3} + 99 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 226 T^{2} + 22283 T^{4} - 1239588 T^{6} + 22283 p^{2} T^{8} - 226 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 226 T^{2} + 23519 T^{4} - 1415868 T^{6} + 23519 p^{2} T^{8} - 226 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 150 T^{2} + 6827 T^{4} - 178700 T^{6} + 6827 p^{2} T^{8} - 150 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 - 10 T + 89 T^{2} - 332 T^{3} + 89 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 - 10 T + 137 T^{2} - 936 T^{3} + 137 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 210 T^{2} + 23675 T^{4} - 1807940 T^{6} + 23675 p^{2} T^{8} - 210 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 + 10 T + 149 T^{2} + 908 T^{3} + 149 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 110 T^{2} + 7471 T^{4} - 434276 T^{6} + 7471 p^{2} T^{8} - 110 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 - 4 T + 145 T^{2} - 280 T^{3} + 145 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 + 50 T^{2} + 8667 T^{4} + 476900 T^{6} + 8667 p^{2} T^{8} + 50 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 - 16 T + 275 T^{2} - 2336 T^{3} + 275 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 218 T^{2} + 7375 T^{4} + 853972 T^{6} + 7375 p^{2} T^{8} - 218 p^{4} T^{10} + p^{6} T^{12} \) |
| 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
−5.64737659248722745051044075774, −5.40591622794034863689529574207, −5.16589250033980590372505808274, −5.07255632868025659198689828871, −4.97107265870888289876372274350, −4.71510415517577286046724848139, −4.51929739412161778100429602754, −4.31987721145342153275099844166, −4.22864468195291027647654693924, −4.17573658545271689436779512133, −3.82924925713419862711075448139, −3.77289180288822858883669352432, −3.47090948855882169526379586761, −3.39341174655318573982580196611, −2.98808823592940161702850120845, −2.74262348727467866646447346260, −2.69505518903088712497815594950, −2.48500241627884466602472658137, −2.17794977545268323621627770215, −2.00437424204965669508412646112, −1.86111990114799907894375734410, −1.04877170522154776089853271380, −1.01915319468496588813221308128, −0.62559278839599645733217457335, −0.49791651549522909764257846772,
0.49791651549522909764257846772, 0.62559278839599645733217457335, 1.01915319468496588813221308128, 1.04877170522154776089853271380, 1.86111990114799907894375734410, 2.00437424204965669508412646112, 2.17794977545268323621627770215, 2.48500241627884466602472658137, 2.69505518903088712497815594950, 2.74262348727467866646447346260, 2.98808823592940161702850120845, 3.39341174655318573982580196611, 3.47090948855882169526379586761, 3.77289180288822858883669352432, 3.82924925713419862711075448139, 4.17573658545271689436779512133, 4.22864468195291027647654693924, 4.31987721145342153275099844166, 4.51929739412161778100429602754, 4.71510415517577286046724848139, 4.97107265870888289876372274350, 5.07255632868025659198689828871, 5.16589250033980590372505808274, 5.40591622794034863689529574207, 5.64737659248722745051044075774
Plot not available for L-functions of degree greater than 10.
