| L(s) = 1 | + 2·4-s − 5-s + 2·9-s + 2·11-s − 3·16-s − 2·20-s + 2·25-s + 36·29-s + 4·36-s − 12·41-s + 4·44-s − 2·45-s + 23·49-s − 2·55-s + 20·59-s − 14·61-s − 8·64-s − 52·71-s − 24·79-s + 3·80-s + 81-s + 24·89-s + 4·99-s + 4·100-s + 20·101-s + 40·109-s + 72·116-s + ⋯ |
| L(s) = 1 | + 4-s − 0.447·5-s + 2/3·9-s + 0.603·11-s − 3/4·16-s − 0.447·20-s + 2/5·25-s + 6.68·29-s + 2/3·36-s − 1.87·41-s + 0.603·44-s − 0.298·45-s + 23/7·49-s − 0.269·55-s + 2.60·59-s − 1.79·61-s − 64-s − 6.17·71-s − 2.70·79-s + 0.335·80-s + 1/9·81-s + 2.54·89-s + 0.402·99-s + 2/5·100-s + 1.99·101-s + 3.83·109-s + 6.68·116-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(13.76912397\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.76912397\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T - T^{2} - 2 T^{3} - p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - p T^{2} + 7 T^{4} - 3 p^{2} T^{6} + 7 p^{2} T^{8} - p^{5} T^{10} + p^{6} T^{12} \) |
| 3 | \( 1 - 2 T^{2} + p T^{4} - 20 T^{6} + p^{3} T^{8} - 2 p^{4} T^{10} + p^{6} T^{12} \) |
| 7 | \( 1 - 23 T^{2} + 307 T^{4} - 2586 T^{6} + 307 p^{2} T^{8} - 23 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 - T + 17 T^{2} - 10 T^{3} + 17 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 - 50 T^{2} + 1315 T^{4} - 21108 T^{6} + 1315 p^{2} T^{8} - 50 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( ( 1 - 9 T + 19 T^{2} + 18 T^{3} + 19 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )( 1 + 9 T + 19 T^{2} - 18 T^{3} + 19 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} ) \) |
| 23 | \( 1 - 102 T^{2} + 4831 T^{4} - 138580 T^{6} + 4831 p^{2} T^{8} - 102 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 - 6 T + p T^{2} )^{6} \) |
| 31 | \( ( 1 + 37 T^{2} - 128 T^{3} + 37 p T^{4} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 166 T^{2} + 13011 T^{4} - 608316 T^{6} + 13011 p^{2} T^{8} - 166 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 + 6 T + 79 T^{2} + 516 T^{3} + 79 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 239 T^{2} + 24571 T^{4} - 1388154 T^{6} + 24571 p^{2} T^{8} - 239 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 95 T^{2} + 5443 T^{4} - 214314 T^{6} + 5443 p^{2} T^{8} - 95 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 162 T^{2} + 11539 T^{4} - 610708 T^{6} + 11539 p^{2} T^{8} - 162 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 - 10 T + 185 T^{2} - 1132 T^{3} + 185 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 + 7 T + 79 T^{2} + 78 T^{3} + 79 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 62 T^{2} + 4771 T^{4} - 199788 T^{6} + 4771 p^{2} T^{8} - 62 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 + 26 T + 413 T^{2} + 4124 T^{3} + 413 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 307 T^{2} + 43299 T^{4} - 3822498 T^{6} + 43299 p^{2} T^{8} - 307 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 12 T + 229 T^{2} + 1864 T^{3} + 229 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 270 T^{2} + 39367 T^{4} - 3817060 T^{6} + 39367 p^{2} T^{8} - 270 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 - 12 T - 17 T^{2} + 1320 T^{3} - 17 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 554 T^{2} + 130507 T^{4} - 16717956 T^{6} + 130507 p^{2} T^{8} - 554 p^{4} T^{10} + p^{6} T^{12} \) |
| 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
−4.76851183913620116152113485899, −4.61693235331069114780833830615, −4.48504821491565818703401380171, −4.36095544569600037484058357968, −4.34641720510315174858328721056, −4.19036253732449386584103905424, −4.08474769651222406474469935681, −3.64631606416388633308605903292, −3.53230361095427239104790116697, −3.40586582223067346002101311999, −3.27046501499706142015467341929, −2.93365441519194347555078878973, −2.85632215634603786780555926587, −2.72044275827066815376355793963, −2.65973443003436196141664137675, −2.46054041348158888457180397412, −2.24274121837957531325646395571, −2.05448114651725341620486809027, −1.61099698428263238427734397148, −1.49510889087066935215285968382, −1.43468207357743449761754017209, −1.23957675572701137288802427525, −0.65602972066076917665179607763, −0.58369764332420719095883884932, −0.57597174279959221466645074830,
0.57597174279959221466645074830, 0.58369764332420719095883884932, 0.65602972066076917665179607763, 1.23957675572701137288802427525, 1.43468207357743449761754017209, 1.49510889087066935215285968382, 1.61099698428263238427734397148, 2.05448114651725341620486809027, 2.24274121837957531325646395571, 2.46054041348158888457180397412, 2.65973443003436196141664137675, 2.72044275827066815376355793963, 2.85632215634603786780555926587, 2.93365441519194347555078878973, 3.27046501499706142015467341929, 3.40586582223067346002101311999, 3.53230361095427239104790116697, 3.64631606416388633308605903292, 4.08474769651222406474469935681, 4.19036253732449386584103905424, 4.34641720510315174858328721056, 4.36095544569600037484058357968, 4.48504821491565818703401380171, 4.61693235331069114780833830615, 4.76851183913620116152113485899
Plot not available for L-functions of degree greater than 10.
