L(s) = 1 | + 2·2-s − 2·3-s − 3·4-s + 3·5-s − 4·6-s − 6·7-s − 10·8-s + 7·9-s + 6·10-s − 6·11-s + 6·12-s + 3·13-s − 12·14-s − 6·15-s − 4·17-s + 14·18-s − 10·19-s − 9·20-s + 12·21-s − 12·22-s + 4·23-s + 20·24-s + 11·25-s + 6·26-s − 22·27-s + 18·28-s + 30·29-s + ⋯ |
L(s) = 1 | + 1.41·2-s − 1.15·3-s − 3/2·4-s + 1.34·5-s − 1.63·6-s − 2.26·7-s − 3.53·8-s + 7/3·9-s + 1.89·10-s − 1.80·11-s + 1.73·12-s + 0.832·13-s − 3.20·14-s − 1.54·15-s − 0.970·17-s + 3.29·18-s − 2.29·19-s − 2.01·20-s + 2.61·21-s − 2.55·22-s + 0.834·23-s + 4.08·24-s + 11/5·25-s + 1.17·26-s − 4.23·27-s + 3.40·28-s + 5.57·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(31^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7917420318\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7917420318\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 31 | | \( 1 \) |
good | 2 | $D_{4}$ | \( ( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 3 | $C_2^2$ | \( ( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 5 | $D_4\times C_2$ | \( 1 - 3 T - 2 T^{2} - 3 T^{3} + 51 T^{4} - 3 p T^{5} - 2 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_4\times C_2$ | \( 1 + 6 T + 10 T^{2} + 24 T^{3} + 159 T^{4} + 24 p T^{5} + 10 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 3 T - 8 T^{2} + 27 T^{3} + 3 T^{4} + 27 p T^{5} - 8 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 4 T - p T^{2} - 4 T^{3} + 528 T^{4} - 4 p T^{5} - p^{3} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 - 15 T + 113 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 4 T - 57 T^{2} - 4 T^{3} + 3368 T^{4} - 4 p T^{5} - 57 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 4 T - 50 T^{2} - 64 T^{3} + 2019 T^{4} - 64 p T^{5} - 50 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 7 T - 48 T^{2} + 77 T^{3} + 5453 T^{4} + 77 p T^{5} - 48 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 9 T + 113 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 + 12 T + 47 T^{2} - 108 T^{3} - 1032 T^{4} - 108 p T^{5} + 47 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 10 T - 23 T^{2} + 50 T^{3} + 5748 T^{4} + 50 p T^{5} - 23 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 4 T - 117 T^{2} + 4 T^{3} + 12128 T^{4} + 4 p T^{5} - 117 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 11 T - 20 T^{2} + 11 T^{3} + 6249 T^{4} + 11 p T^{5} - 20 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 3 T - 38 T^{2} + 297 T^{3} - 3777 T^{4} + 297 p T^{5} - 38 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 3 T - 128 T^{2} + 87 T^{3} + 11133 T^{4} + 87 p T^{5} - 128 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 15 T + 233 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 24 T + 293 T^{2} + 24 p T^{3} + p^{2} T^{4} )^{2} \) |
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
−6.90403984327971351532389130364, −6.88271757958898352983330730347, −6.39751679491253939192114953790, −6.31364475381800771740623459203, −6.28992385563979497663963529521, −6.15986399964514616504511065712, −5.95097819380783799009366551168, −5.21810411800881077048588835710, −5.19845117076776146872257342730, −5.18183032494642023202372265547, −4.93285881413882121508174583192, −4.51341724400154328630822860052, −4.45474619958065899136456612759, −4.30972077593198420737730185885, −4.25753568610799782494438246153, −3.43428919097978450756725829672, −3.28028795792254990178047517501, −3.26764808434056377771295193647, −3.09638441642875175622430376000, −2.41800467465297864579022774625, −2.19086084405425483947862188507, −1.81520953745442871453278216255, −1.23509293784862175622206171081, −0.61312467116054062893572141416, −0.30269298151513855362829477622,
0.30269298151513855362829477622, 0.61312467116054062893572141416, 1.23509293784862175622206171081, 1.81520953745442871453278216255, 2.19086084405425483947862188507, 2.41800467465297864579022774625, 3.09638441642875175622430376000, 3.26764808434056377771295193647, 3.28028795792254990178047517501, 3.43428919097978450756725829672, 4.25753568610799782494438246153, 4.30972077593198420737730185885, 4.45474619958065899136456612759, 4.51341724400154328630822860052, 4.93285881413882121508174583192, 5.18183032494642023202372265547, 5.19845117076776146872257342730, 5.21810411800881077048588835710, 5.95097819380783799009366551168, 6.15986399964514616504511065712, 6.28992385563979497663963529521, 6.31364475381800771740623459203, 6.39751679491253939192114953790, 6.88271757958898352983330730347, 6.90403984327971351532389130364