| L(s) = 1 | + 4·5-s − 4·7-s − 2·9-s + 4·11-s − 8·13-s − 8·16-s + 8·17-s − 8·19-s + 8·25-s + 20·29-s + 16·31-s − 16·35-s − 8·37-s + 16·41-s − 8·45-s + 20·47-s + 8·49-s − 4·53-s + 16·55-s + 8·59-s + 8·63-s − 32·65-s + 24·71-s − 16·73-s − 16·77-s − 20·79-s − 32·80-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 1.51·7-s − 2/3·9-s + 1.20·11-s − 2.21·13-s − 2·16-s + 1.94·17-s − 1.83·19-s + 8/5·25-s + 3.71·29-s + 2.87·31-s − 2.70·35-s − 1.31·37-s + 2.49·41-s − 1.19·45-s + 2.91·47-s + 8/7·49-s − 0.549·53-s + 2.15·55-s + 1.04·59-s + 1.00·63-s − 3.96·65-s + 2.84·71-s − 1.87·73-s − 1.82·77-s − 2.25·79-s − 3.57·80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.939926291\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.939926291\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2}( 1 + p T + p T^{2} )^{2} \) |
| 5 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 8 T^{3} - T^{4} - 8 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 - 18 T^{2} + p^{2} T^{4} ) \) |
| 17 | $D_{4}$ | \( ( 1 - 4 T + 28 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 176 T^{3} + 959 T^{4} + 176 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 70 T^{2} + 2243 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 10 T + 73 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 928 T^{3} + 5999 T^{4} - 928 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} - 280 T^{3} - 2734 T^{4} - 280 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 848 T^{3} + 5474 T^{4} - 848 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 90 T^{2} + 5563 T^{4} - 90 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 20 T + 200 T^{2} - 1840 T^{3} + 14903 T^{4} - 1840 p T^{5} + 200 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 2 T + 97 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} + 104 T^{3} - 4846 T^{4} + 104 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 112 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 - 6062 T^{4} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 1600 T^{3} + 19271 T^{4} + 1600 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 10 T + 143 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 20 T + 200 T^{2} - 2560 T^{3} + 30743 T^{4} - 2560 p T^{5} + 200 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 28 T + 392 T^{2} - 5096 T^{3} + 57599 T^{4} - 5096 p T^{5} + 392 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 800 T^{3} + 19991 T^{4} - 800 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 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
−8.811081513300056959973063548806, −8.550246508769274160247298944389, −8.280920157273992439576271577395, −7.999279660337722590277705734908, −7.43223274598927216856801252440, −7.29007512988292688393565735336, −7.13620835045515447514980192708, −6.48839325982622454478528687318, −6.44613244189136516747984825918, −6.34328307374327436666686718148, −6.33448984027236089576509141357, −5.87011274459756255428959965400, −5.48536526890586633882438120054, −5.04337118978101835240087256500, −4.83358061923825864662554478684, −4.72041552179378672385096697979, −4.19875096733809641258055966341, −3.93241317483145620196418602690, −3.53601177528004863786543362319, −2.68649122862001342873151825280, −2.63153434223661009255706516467, −2.50501054311055669118991620702, −2.40223591361940046914155219368, −1.30392652492862420512160130643, −0.73515486922434665240353910401,
0.73515486922434665240353910401, 1.30392652492862420512160130643, 2.40223591361940046914155219368, 2.50501054311055669118991620702, 2.63153434223661009255706516467, 2.68649122862001342873151825280, 3.53601177528004863786543362319, 3.93241317483145620196418602690, 4.19875096733809641258055966341, 4.72041552179378672385096697979, 4.83358061923825864662554478684, 5.04337118978101835240087256500, 5.48536526890586633882438120054, 5.87011274459756255428959965400, 6.33448984027236089576509141357, 6.34328307374327436666686718148, 6.44613244189136516747984825918, 6.48839325982622454478528687318, 7.13620835045515447514980192708, 7.29007512988292688393565735336, 7.43223274598927216856801252440, 7.999279660337722590277705734908, 8.280920157273992439576271577395, 8.550246508769274160247298944389, 8.811081513300056959973063548806
