| L(s) = 1 | − 2·2-s + 4·3-s + 4-s − 4·5-s − 8·6-s − 4·7-s + 2·8-s − 2·9-s + 8·10-s + 4·12-s − 4·13-s + 8·14-s − 16·15-s − 4·16-s + 17-s + 4·18-s − 3·19-s − 4·20-s − 16·21-s + 8·24-s + 10·25-s + 8·26-s − 40·27-s − 4·28-s − 12·29-s + 32·30-s + 31-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 2.30·3-s + 1/2·4-s − 1.78·5-s − 3.26·6-s − 1.51·7-s + 0.707·8-s − 2/3·9-s + 2.52·10-s + 1.15·12-s − 1.10·13-s + 2.13·14-s − 4.13·15-s − 16-s + 0.242·17-s + 0.942·18-s − 0.688·19-s − 0.894·20-s − 3.49·21-s + 1.63·24-s + 2·25-s + 1.56·26-s − 7.69·27-s − 0.755·28-s − 2.22·29-s + 5.84·30-s + 0.179·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 67^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 67^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1217844112\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1217844112\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{4} \) |
| 67 | $C_2^2$ | \( 1 + 15 T + 142 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 7 | $C_2^2$ | \( ( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - T + 15 T^{2} + 48 T^{3} - 110 T^{4} + 48 p T^{5} + 15 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 3 T + 17 T^{2} - 138 T^{3} - 468 T^{4} - 138 p T^{5} + 17 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - T - 13 T^{2} + 48 T^{3} - 796 T^{4} + 48 p T^{5} - 13 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 3 T - 19 T^{2} + 138 T^{3} - 918 T^{4} + 138 p T^{5} - 19 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 5 T - 15 T^{2} - 210 T^{3} - 1106 T^{4} - 210 p T^{5} - 15 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 - 5 T + 64 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_{4}$ | \( ( 1 - 13 T + 112 T^{2} - 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 + 5 T - 75 T^{2} - 210 T^{3} + 3184 T^{4} - 210 p T^{5} - 75 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 + 17 T + 119 T^{2} + 408 T^{3} + 2474 T^{4} + 408 p T^{5} + 119 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 7 T - 81 T^{2} + 252 T^{3} + 5944 T^{4} + 252 p T^{5} - 81 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 4 T - 11 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 5 T - 127 T^{2} - 210 T^{3} + 11270 T^{4} - 210 p T^{5} - 127 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 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
−7.58232544266138531500585877531, −7.55937840411831058354189560162, −7.45970636448954167972164566535, −7.28980335435967299213476191099, −6.73922398861450374116841939217, −6.59771370604837449373916304024, −6.13985326694211103858299743254, −5.94345582787486563911664055062, −5.80057318961531591949219658152, −5.38760884581231267846360778791, −5.20211395523975798788328468998, −4.90033498893668724580348561983, −4.32120361969486359156557723068, −4.30215732299665809351944490528, −3.72325213058338458500796496860, −3.57227165549340172204953883704, −3.56999249127460899495311542879, −3.14111112712581098015872382262, −2.92873602934317063659094526084, −2.59723595317743128717502856547, −2.47388025158436851477249216373, −1.93594985545637316623532541233, −1.69340939986965152604854819180, −0.49211392820243091364537608345, −0.20531021329485706268201773127,
0.20531021329485706268201773127, 0.49211392820243091364537608345, 1.69340939986965152604854819180, 1.93594985545637316623532541233, 2.47388025158436851477249216373, 2.59723595317743128717502856547, 2.92873602934317063659094526084, 3.14111112712581098015872382262, 3.56999249127460899495311542879, 3.57227165549340172204953883704, 3.72325213058338458500796496860, 4.30215732299665809351944490528, 4.32120361969486359156557723068, 4.90033498893668724580348561983, 5.20211395523975798788328468998, 5.38760884581231267846360778791, 5.80057318961531591949219658152, 5.94345582787486563911664055062, 6.13985326694211103858299743254, 6.59771370604837449373916304024, 6.73922398861450374116841939217, 7.28980335435967299213476191099, 7.45970636448954167972164566535, 7.55937840411831058354189560162, 7.58232544266138531500585877531
