| L(s) = 1 | − 3-s + 2·4-s + 5·5-s + 6·7-s + 11·11-s − 2·12-s + 6·13-s − 5·15-s + 17-s − 13·19-s + 10·20-s − 6·21-s − 23-s + 10·25-s + 12·28-s − 14·29-s − 8·31-s − 11·33-s + 30·35-s + 13·37-s − 6·39-s + 16·41-s + 16·43-s + 22·44-s + 27·49-s − 51-s + 12·52-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 4-s + 2.23·5-s + 2.26·7-s + 3.31·11-s − 0.577·12-s + 1.66·13-s − 1.29·15-s + 0.242·17-s − 2.98·19-s + 2.23·20-s − 1.30·21-s − 0.208·23-s + 2·25-s + 2.26·28-s − 2.59·29-s − 1.43·31-s − 1.91·33-s + 5.07·35-s + 2.13·37-s − 0.960·39-s + 2.49·41-s + 2.43·43-s + 3.31·44-s + 27/7·49-s − 0.140·51-s + 1.66·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.57931161\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.57931161\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 5 | $C_4$ | \( 1 - p T + 3 p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 - p T + 51 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| good | 2 | $C_4\times C_2$ | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2^2:C_4$ | \( 1 - 6 T + 9 T^{2} + 38 T^{3} - 201 T^{4} + 38 p T^{5} + 9 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 - 6 T + 3 T^{2} + 80 T^{3} - 399 T^{4} + 80 p T^{5} + 3 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 - T + 14 T^{2} - 37 T^{3} + 359 T^{4} - 37 p T^{5} + 14 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 + 13 T + 50 T^{2} - 37 T^{3} - 791 T^{4} - 37 p T^{5} + 50 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2:C_4$ | \( 1 + T - 7 T^{2} - 95 T^{3} + 236 T^{4} - 95 p T^{5} - 7 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2:C_4$ | \( 1 + 14 T + 107 T^{2} + 762 T^{3} + 4865 T^{4} + 762 p T^{5} + 107 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 37 | $C_2^2:C_4$ | \( 1 - 13 T + 42 T^{2} + 415 T^{3} - 5029 T^{4} + 415 p T^{5} + 42 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 8 T + 93 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 8 T + 97 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 - 37 T^{2} - 210 T^{3} + 1999 T^{4} - 210 p T^{5} - 37 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_4\times C_2$ | \( 1 + 9 T + 28 T^{2} - 225 T^{3} - 3509 T^{4} - 225 p T^{5} + 28 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 - 15 T + 56 T^{2} + 705 T^{3} - 10799 T^{4} + 705 p T^{5} + 56 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 - 15 T + 204 T^{2} - 1985 T^{3} + 19251 T^{4} - 1985 p T^{5} + 204 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_4\times C_2$ | \( 1 + 5 T - 52 T^{2} + 25 T^{3} + 4849 T^{4} + 25 p T^{5} - 52 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 14 T + 171 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 2 T + 102 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2:C_4$ | \( 1 - 19 T + 57 T^{2} + 1093 T^{3} - 14020 T^{4} + 1093 p T^{5} + 57 p^{2} T^{6} - 19 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 + 19 T + 53 T^{2} - 1145 T^{3} - 14404 T^{4} - 1145 p T^{5} + 53 p^{2} T^{6} + 19 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2^2:C_4$ | \( 1 + 21 T + 82 T^{2} - 987 T^{3} - 13625 T^{4} - 987 p T^{5} + 82 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^2:C_4$ | \( 1 - 8 T - 63 T^{2} + 980 T^{3} - 1099 T^{4} + 980 p T^{5} - 63 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 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
−7.39417561474133248529222084927, −6.88431184045035407925439040231, −6.77368049153225178107741899994, −6.64277381519481764208462653994, −6.31507758210817470137496155843, −6.02026353347847625093688796388, −5.95259097990392430900725268471, −5.90594442629605209396769151066, −5.71647227158032460881386931406, −5.33660453685831254739805021954, −5.24702642586929971226439688045, −4.59646022726115509438449553417, −4.41835457874730489236940159864, −4.04190503697273249870063494998, −3.97495364204366857557835664364, −3.91350562451302433692976383094, −3.80284173617101765334024278814, −2.69945627136653135774003699166, −2.60902084660006798921724711411, −2.37977880753265267237120497144, −1.92842095892573699555912674571, −1.72892718357497994306824864663, −1.52457450253782426516071951460, −1.31297342379147215333393254848, −0.856634161752112867634522934186,
0.856634161752112867634522934186, 1.31297342379147215333393254848, 1.52457450253782426516071951460, 1.72892718357497994306824864663, 1.92842095892573699555912674571, 2.37977880753265267237120497144, 2.60902084660006798921724711411, 2.69945627136653135774003699166, 3.80284173617101765334024278814, 3.91350562451302433692976383094, 3.97495364204366857557835664364, 4.04190503697273249870063494998, 4.41835457874730489236940159864, 4.59646022726115509438449553417, 5.24702642586929971226439688045, 5.33660453685831254739805021954, 5.71647227158032460881386931406, 5.90594442629605209396769151066, 5.95259097990392430900725268471, 6.02026353347847625093688796388, 6.31507758210817470137496155843, 6.64277381519481764208462653994, 6.77368049153225178107741899994, 6.88431184045035407925439040231, 7.39417561474133248529222084927
