| L(s) = 1 | + 4·4-s + 12·16-s + 112·23-s + 80·25-s + 112·29-s + 64·37-s − 128·43-s − 168·53-s + 32·64-s + 16·67-s + 448·71-s − 144·79-s + 448·92-s + 320·100-s + 112·107-s + 32·109-s − 56·113-s + 448·116-s − 468·121-s + 127-s + 131-s + 137-s + 139-s + 256·148-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | + 4-s + 3/4·16-s + 4.86·23-s + 16/5·25-s + 3.86·29-s + 1.72·37-s − 2.97·43-s − 3.16·53-s + 1/2·64-s + 0.238·67-s + 6.30·71-s − 1.82·79-s + 4.86·92-s + 16/5·100-s + 1.04·107-s + 0.293·109-s − 0.495·113-s + 3.86·116-s − 3.86·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 1.72·148-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(12.67222916\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.67222916\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2:C_4$ | \( 1 - 16 p T^{2} + 2752 T^{4} - 16 p^{5} T^{6} + p^{8} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 234 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2:C_4$ | \( 1 - 320 T^{2} + 54400 T^{4} - 320 p^{4} T^{6} + p^{8} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 - 688 T^{2} + 960 p^{2} T^{4} - 688 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 - 900 T^{2} + 456870 T^{4} - 900 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 56 T + 1770 T^{2} - 56 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 - 56 T + 2224 T^{2} - 56 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2:C_4$ | \( 1 - 2276 T^{2} + 2834758 T^{4} - 2276 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 32 T + 2112 T^{2} - 32 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2:C_4$ | \( 1 - 1040 T^{2} + 3520 p^{2} T^{4} - 1040 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 64 T + 3154 T^{2} + 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 + 1564 T^{2} + 6450886 T^{4} + 1564 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 84 T + 3510 T^{2} + 84 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2:C_4$ | \( 1 - 5316 T^{2} + 17444838 T^{4} - 5316 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 - 9152 T^{2} + 41434240 T^{4} - 9152 p^{4} T^{6} + p^{8} T^{8} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p^{2} T^{2} )^{4} \) |
| 71 | $D_{4}$ | \( ( 1 - 224 T + 22234 T^{2} - 224 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2:C_4$ | \( 1 - 12864 T^{2} + 88688448 T^{4} - 12864 p^{4} T^{6} + p^{8} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 72 T + 12210 T^{2} + 72 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2:C_4$ | \( 1 - 3876 T^{2} + 69670758 T^{4} - 3876 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $C_2^2:C_4$ | \( 1 - 6768 T^{2} + 136931136 T^{4} - 6768 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $C_2^2:C_4$ | \( 1 - 15168 T^{2} + 120056640 T^{4} - 15168 p^{4} T^{6} + p^{8} T^{8} \) |
| 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
−6.91887295989015821618023567686, −6.73649062666131920809098873022, −6.64867928188341954428571935806, −6.61286725052009565089526291812, −6.40247872378090919574245742756, −6.05918785485257066675231171958, −5.64227000714808811679346990504, −5.20096923423759889207768243566, −5.16635917511823576150251238689, −4.92762998568177963923305362654, −4.76984678087627624228398029246, −4.56660200105487468940118985797, −4.55571265224317475181355686730, −3.74692380350638999830950218328, −3.36827771458288660250953077378, −3.36049015038990907232615518406, −3.07410455198320801238847409696, −2.68986230021486152592500137760, −2.58754575948225112483268231327, −2.58506199453950689293508654985, −1.78110355783113400873418940804, −1.25457825094413872687047287731, −1.12132898941795147926253189414, −0.949812879229394690609727890767, −0.55422460918542697101666641065,
0.55422460918542697101666641065, 0.949812879229394690609727890767, 1.12132898941795147926253189414, 1.25457825094413872687047287731, 1.78110355783113400873418940804, 2.58506199453950689293508654985, 2.58754575948225112483268231327, 2.68986230021486152592500137760, 3.07410455198320801238847409696, 3.36049015038990907232615518406, 3.36827771458288660250953077378, 3.74692380350638999830950218328, 4.55571265224317475181355686730, 4.56660200105487468940118985797, 4.76984678087627624228398029246, 4.92762998568177963923305362654, 5.16635917511823576150251238689, 5.20096923423759889207768243566, 5.64227000714808811679346990504, 6.05918785485257066675231171958, 6.40247872378090919574245742756, 6.61286725052009565089526291812, 6.64867928188341954428571935806, 6.73649062666131920809098873022, 6.91887295989015821618023567686
