| L(s) = 1 | + 2·4-s + 26·7-s + 60·13-s + 26·19-s − 48·25-s + 52·28-s − 6·31-s − 34·37-s − 340·43-s + 409·49-s + 120·52-s + 144·61-s − 8·64-s − 86·67-s + 190·73-s + 52·76-s − 138·79-s + 1.56e3·91-s + 64·97-s − 96·100-s + 122·103-s + 130·109-s − 192·121-s − 12·124-s + 127-s + 131-s + 676·133-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 26/7·7-s + 4.61·13-s + 1.36·19-s − 1.91·25-s + 13/7·28-s − 0.193·31-s − 0.918·37-s − 7.90·43-s + 8.34·49-s + 2.30·52-s + 2.36·61-s − 1/8·64-s − 1.28·67-s + 2.60·73-s + 0.684·76-s − 1.74·79-s + 17.1·91-s + 0.659·97-s − 0.959·100-s + 1.18·103-s + 1.19·109-s − 1.58·121-s − 0.0967·124-s + 0.00787·127-s + 0.00763·131-s + 5.08·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(5.990667707\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.990667707\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - 13 T + p^{2} T^{2} )^{2} \) |
| good | 5 | $C_2^3$ | \( 1 + 48 T^{2} + 1679 T^{4} + 48 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $C_2^3$ | \( 1 + 192 T^{2} + 22223 T^{4} + 192 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - 15 T + p^{2} T^{2} )^{4} \) |
| 17 | $C_2^3$ | \( 1 + 450 T^{2} + 118979 T^{4} + 450 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 13 T - 192 T^{2} - 13 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 546 T^{2} + 18275 T^{4} + 546 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 1170 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 3 T - 952 T^{2} + 3 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 17 T - 1080 T^{2} + 17 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 3136 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 85 T + p^{2} T^{2} )^{4} \) |
| 47 | $C_2^3$ | \( 1 - 784 T^{2} - 4265025 T^{4} - 784 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 4466 T^{2} + 12054675 T^{4} + 4466 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $C_2^3$ | \( 1 - 1230 T^{2} - 10604461 T^{4} - 1230 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 72 T + 1463 T^{2} - 72 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 43 T - 2640 T^{2} + 43 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 7344 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 95 T + 3696 T^{2} - 95 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 69 T - 1480 T^{2} + 69 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 10080 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - 2590 T^{2} - 56034141 T^{4} - 2590 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )^{4} \) |
| 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
−9.886456033873053057547698701440, −8.781042143002307097180148557661, −8.756887335646631228562842770883, −8.739241601382243938514227346113, −8.722888319660509396132606708941, −8.040117888735218415860618520652, −7.900159430104393461899120165310, −7.87632734572414259186465954464, −7.57403421057571864717543549380, −6.82554066571279616578355568653, −6.60099715613805147644968038905, −6.47383219470546976527585392406, −6.03456938735435264446836970504, −5.44626209785701800683384994054, −5.37960733032248306767664161930, −5.01519337849101024268367096265, −4.94199475222599745543962838668, −4.11551439767430689649463932261, −3.83648397381795246772166286568, −3.54997235411080694041214942821, −3.35453244028767679761011196441, −2.23655606928135104505571122569, −1.61804728909872112504619206933, −1.44299458482112654557340022845, −1.35932590013401162595765589110,
1.35932590013401162595765589110, 1.44299458482112654557340022845, 1.61804728909872112504619206933, 2.23655606928135104505571122569, 3.35453244028767679761011196441, 3.54997235411080694041214942821, 3.83648397381795246772166286568, 4.11551439767430689649463932261, 4.94199475222599745543962838668, 5.01519337849101024268367096265, 5.37960733032248306767664161930, 5.44626209785701800683384994054, 6.03456938735435264446836970504, 6.47383219470546976527585392406, 6.60099715613805147644968038905, 6.82554066571279616578355568653, 7.57403421057571864717543549380, 7.87632734572414259186465954464, 7.900159430104393461899120165310, 8.040117888735218415860618520652, 8.722888319660509396132606708941, 8.739241601382243938514227346113, 8.756887335646631228562842770883, 8.781042143002307097180148557661, 9.886456033873053057547698701440
