| L(s) = 1 | + 6·4-s + 4·5-s + 8·11-s + 19·16-s + 8·19-s + 24·20-s + 2·25-s − 4·29-s + 8·31-s − 8·41-s + 48·44-s + 12·49-s + 32·55-s − 16·59-s + 24·61-s + 36·64-s − 16·71-s + 48·76-s + 24·79-s + 76·80-s − 24·89-s + 32·95-s + 12·100-s − 24·101-s + 8·109-s − 24·116-s + 12·121-s + ⋯ |
| L(s) = 1 | + 3·4-s + 1.78·5-s + 2.41·11-s + 19/4·16-s + 1.83·19-s + 5.36·20-s + 2/5·25-s − 0.742·29-s + 1.43·31-s − 1.24·41-s + 7.23·44-s + 12/7·49-s + 4.31·55-s − 2.08·59-s + 3.07·61-s + 9/2·64-s − 1.89·71-s + 5.50·76-s + 2.70·79-s + 8.49·80-s − 2.54·89-s + 3.28·95-s + 6/5·100-s − 2.38·101-s + 0.766·109-s − 2.22·116-s + 1.09·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(22.76975928\) |
| \(L(\frac12)\) |
\(\approx\) |
\(22.76975928\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 29 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 2 | $C_2^2$ | \( ( 1 - 3 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $D_{4}$ | \( ( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_4\times C_2$ | \( 1 + 4 T^{2} + 70 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 4 T + 34 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 4 T + 58 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 3094 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 61 | $D_{4}$ | \( ( 1 - 12 T + 126 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 220 T^{2} + 20566 T^{4} - 220 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 220 T^{2} + 22246 T^{4} - 220 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 12 T + 186 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 156 T^{2} + 15254 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 12 T + 182 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 252 T^{2} + 30086 T^{4} - 252 p^{2} T^{6} + p^{4} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.75653805368517480898485240807, −6.72590487525942106592720369806, −6.47251972729842819391816532071, −6.17222462254800643516582272826, −6.05791286036825512557174663435, −5.95444276311360181418156089799, −5.90066105930186670856888943059, −5.33475789557022664169175653492, −5.27965698521830647414869971855, −5.00354350830600644692119189111, −4.90919780879598798610188529546, −4.09069885793520338764019605144, −4.08924836792444133224851486239, −3.87862340877025694566827492507, −3.68852687533711449472520610048, −3.20233020249457484227067844533, −2.81437949568603745256367102200, −2.74548429894043331506252831058, −2.73606759216184989290284277598, −2.16666218826193943579253681620, −1.74831184907352485757634631391, −1.66231757131814150783198163943, −1.65828633201886436311470182555, −1.10527983121421442494362570897, −0.824903597857932196047040992753,
0.824903597857932196047040992753, 1.10527983121421442494362570897, 1.65828633201886436311470182555, 1.66231757131814150783198163943, 1.74831184907352485757634631391, 2.16666218826193943579253681620, 2.73606759216184989290284277598, 2.74548429894043331506252831058, 2.81437949568603745256367102200, 3.20233020249457484227067844533, 3.68852687533711449472520610048, 3.87862340877025694566827492507, 4.08924836792444133224851486239, 4.09069885793520338764019605144, 4.90919780879598798610188529546, 5.00354350830600644692119189111, 5.27965698521830647414869971855, 5.33475789557022664169175653492, 5.90066105930186670856888943059, 5.95444276311360181418156089799, 6.05791286036825512557174663435, 6.17222462254800643516582272826, 6.47251972729842819391816532071, 6.72590487525942106592720369806, 6.75653805368517480898485240807
