| L(s) = 1 | − 88·5-s − 56·13-s + 24·17-s + 3.20e3·25-s + 1.67e3·29-s + 6.31e3·37-s + 4.50e3·41-s + 8.70e3·49-s + 3.33e3·53-s + 1.00e3·61-s + 4.92e3·65-s + 1.73e4·73-s − 2.11e3·85-s + 2.08e4·89-s − 1.17e4·97-s + 2.30e4·101-s − 2.85e4·109-s + 56·113-s + 4.22e4·121-s − 5.20e4·125-s + ⋯ |
| L(s) = 1 | − 3.51·5-s − 0.331·13-s + 0.0830·17-s + 5.12·25-s + 1.98·29-s + 4.61·37-s + 2.67·41-s + 3.62·49-s + 1.18·53-s + 0.268·61-s + 1.16·65-s + 3.25·73-s − 0.292·85-s + 2.63·89-s − 1.25·97-s + 2.25·101-s − 2.40·109-s + 0.00438·113-s + 2.88·121-s − 3.32·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.704599879\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.704599879\) |
| \(L(3)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( ( 1 + 44 T + 1302 T^{2} + 44 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 1244 p T^{2} + 30459270 T^{4} - 1244 p^{9} T^{6} + p^{16} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 42212 T^{2} + 860799366 T^{4} - 42212 p^{8} T^{6} + p^{16} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 28 T + 55590 T^{2} + 28 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 - 12 T - 42010 T^{2} - 12 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 53860 T^{2} + 29953594374 T^{4} - 53860 p^{8} T^{6} + p^{16} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 338948 T^{2} + 84362790 p^{2} T^{4} - 338948 p^{8} T^{6} + p^{16} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 836 T + 1319286 T^{2} - 836 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 497788 T^{2} + 1677652422918 T^{4} + 497788 p^{8} T^{6} + p^{16} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 3156 T + 5070278 T^{2} - 3156 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 2252 T + 6627366 T^{2} - 2252 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 11063780 T^{2} + 52358484947334 T^{4} - 11063780 p^{8} T^{6} + p^{16} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 11767556 T^{2} + 70164493996806 T^{4} - 11767556 p^{8} T^{6} + p^{16} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 1668 T + 7393718 T^{2} - 1668 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 24663908 T^{2} + 392308713389958 T^{4} - 24663908 p^{8} T^{6} + p^{16} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 500 T + 2870982 T^{2} - 500 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 22235108 T^{2} + 484730171092998 T^{4} - 22235108 p^{8} T^{6} + p^{16} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 36810500 T^{2} + 591844540934022 T^{4} - 36810500 p^{8} T^{6} + p^{16} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 8676 T + 75587078 T^{2} - 8676 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 63515012 T^{2} + 2643785888524806 T^{4} - 63515012 p^{8} T^{6} + p^{16} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 77386340 T^{2} + 3588332526206982 T^{4} - 77386340 p^{8} T^{6} + p^{16} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 10428 T + 144203078 T^{2} - 10428 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 5884 T + 158280198 T^{2} + 5884 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
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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
−7.88090375179853735550636451018, −7.59393405609434849774795269339, −7.38896880386533848850574762978, −7.35504354612726801041996562475, −7.27080676012485656526808731573, −6.63943467618970626709451242197, −6.20595712815718877859405542528, −6.10944215104029624562742242759, −6.06242499897248521627199001000, −5.43953927143905791647344911568, −4.92658530786443741435046881944, −4.86628061486365363703164095603, −4.56112294754600947561249377047, −4.02697883065641456414476716579, −3.93993406584542331654185392001, −3.91117093585428592400142359619, −3.77204563220774995221349682365, −2.94619243355833538655473146030, −2.74653285538617737157075705621, −2.41085102964905063299168260946, −2.29482049057242934609955184094, −1.10624719397188711437099228448, −0.808800035726036606574003839613, −0.68748408370326071346901388810, −0.41922745516832755190878698979,
0.41922745516832755190878698979, 0.68748408370326071346901388810, 0.808800035726036606574003839613, 1.10624719397188711437099228448, 2.29482049057242934609955184094, 2.41085102964905063299168260946, 2.74653285538617737157075705621, 2.94619243355833538655473146030, 3.77204563220774995221349682365, 3.91117093585428592400142359619, 3.93993406584542331654185392001, 4.02697883065641456414476716579, 4.56112294754600947561249377047, 4.86628061486365363703164095603, 4.92658530786443741435046881944, 5.43953927143905791647344911568, 6.06242499897248521627199001000, 6.10944215104029624562742242759, 6.20595712815718877859405542528, 6.63943467618970626709451242197, 7.27080676012485656526808731573, 7.35504354612726801041996562475, 7.38896880386533848850574762978, 7.59393405609434849774795269339, 7.88090375179853735550636451018
