| L(s) = 1 | + 3·3-s + 2·4-s + 5-s − 7-s + 3·9-s + 6·12-s − 4·13-s + 3·15-s + 2·17-s − 6·19-s + 2·20-s − 3·21-s − 20·23-s + 5·25-s − 2·28-s + 10·29-s − 31-s − 35-s + 6·36-s + 5·37-s − 12·39-s − 2·41-s + 32·43-s + 3·45-s − 8·47-s + 6·51-s − 8·52-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 4-s + 0.447·5-s − 0.377·7-s + 9-s + 1.73·12-s − 1.10·13-s + 0.774·15-s + 0.485·17-s − 1.37·19-s + 0.447·20-s − 0.654·21-s − 4.17·23-s + 25-s − 0.377·28-s + 1.85·29-s − 0.179·31-s − 0.169·35-s + 36-s + 0.821·37-s − 1.92·39-s − 0.312·41-s + 4.87·43-s + 0.447·45-s − 1.16·47-s + 0.840·51-s − 1.10·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.411449550\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.411449550\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_4\times C_2$ | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 3 | $C_4\times C_2$ | \( 1 - p T + 2 p T^{2} - p^{2} T^{3} + p^{2} T^{4} - p^{3} T^{5} + 2 p^{3} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) |
| 5 | $C_4\times C_2$ | \( 1 - T - 4 T^{2} + 9 T^{3} + 11 T^{4} + 9 p T^{5} - 4 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_4\times C_2$ | \( 1 + 4 T + 3 T^{2} - 40 T^{3} - 199 T^{4} - 40 p T^{5} + 3 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_4\times C_2$ | \( 1 - 2 T - 13 T^{2} + 60 T^{3} + 101 T^{4} + 60 p T^{5} - 13 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_4\times C_2$ | \( 1 + 6 T + 17 T^{2} - 12 T^{3} - 395 T^{4} - 12 p T^{5} + 17 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) |
| 29 | $C_4\times C_2$ | \( 1 - 10 T + 71 T^{2} - 420 T^{3} + 2141 T^{4} - 420 p T^{5} + 71 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_4\times C_2$ | \( 1 + T - 30 T^{2} - 61 T^{3} + 869 T^{4} - 61 p T^{5} - 30 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_4\times C_2$ | \( 1 - 5 T - 12 T^{2} + 245 T^{3} - 781 T^{4} + 245 p T^{5} - 12 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_4\times C_2$ | \( 1 + 2 T - 37 T^{2} - 156 T^{3} + 1205 T^{4} - 156 p T^{5} - 37 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 47 | $C_4\times C_2$ | \( 1 + 8 T + 17 T^{2} - 240 T^{3} - 2719 T^{4} - 240 p T^{5} + 17 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_4\times C_2$ | \( 1 - 6 T - 17 T^{2} + 420 T^{3} - 1619 T^{4} + 420 p T^{5} - 17 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_4\times C_2$ | \( 1 + 3 T - 50 T^{2} - 327 T^{3} + 1969 T^{4} - 327 p T^{5} - 50 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_4\times C_2$ | \( 1 + 2 T - 57 T^{2} - 236 T^{3} + 3005 T^{4} - 236 p T^{5} - 57 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 71 | $C_4\times C_2$ | \( 1 + T - 70 T^{2} - 141 T^{3} + 4829 T^{4} - 141 p T^{5} - 70 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_4\times C_2$ | \( 1 - 10 T + 27 T^{2} + 460 T^{3} - 6571 T^{4} + 460 p T^{5} + 27 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_4\times C_2$ | \( 1 - 6 T - 43 T^{2} + 732 T^{3} - 995 T^{4} + 732 p T^{5} - 43 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_4\times C_2$ | \( 1 - 12 T + 61 T^{2} + 264 T^{3} - 8231 T^{4} + 264 p T^{5} + 61 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{4} \) |
| 97 | $C_4\times C_2$ | \( 1 - 5 T - 72 T^{2} + 845 T^{3} + 2759 T^{4} + 845 p T^{5} - 72 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.45528048174537683151877370280, −7.06043921021673310240955918594, −6.86080151981069932241731810699, −6.76930020858413221706683836030, −6.25218295341912499298566403655, −6.16283522315935989065843124132, −6.10074884962540342783625618715, −5.90845286118685222436989965085, −5.52314628791857521767730749047, −5.26844750815053186609389963032, −4.80913474218338749643048125429, −4.71158784166738734291601555283, −4.27340117459145478232609343687, −3.94173969165015921204476229044, −3.93412662973625388084950185321, −3.87159897017269670202448671487, −3.20103526466802375520514475617, −2.75878029288106046887440356866, −2.60062299094316533223165483521, −2.53766212771330727118091720749, −2.43726768402243278348847815986, −2.05131082770292051708494333216, −1.52456574736774742432399877651, −1.27831561270769580785015426772, −0.34350281464779691706024284713,
0.34350281464779691706024284713, 1.27831561270769580785015426772, 1.52456574736774742432399877651, 2.05131082770292051708494333216, 2.43726768402243278348847815986, 2.53766212771330727118091720749, 2.60062299094316533223165483521, 2.75878029288106046887440356866, 3.20103526466802375520514475617, 3.87159897017269670202448671487, 3.93412662973625388084950185321, 3.94173969165015921204476229044, 4.27340117459145478232609343687, 4.71158784166738734291601555283, 4.80913474218338749643048125429, 5.26844750815053186609389963032, 5.52314628791857521767730749047, 5.90845286118685222436989965085, 6.10074884962540342783625618715, 6.16283522315935989065843124132, 6.25218295341912499298566403655, 6.76930020858413221706683836030, 6.86080151981069932241731810699, 7.06043921021673310240955918594, 7.45528048174537683151877370280
