| L(s) = 1 | − 3·3-s + 2·5-s − 4·7-s + 7·9-s + 10·11-s − 5·13-s − 6·15-s − 17-s + 2·19-s + 12·21-s + 11·23-s + 6·25-s − 18·27-s − 6·29-s − 30·33-s − 8·35-s + 3·37-s + 15·39-s + 20·41-s + 26·43-s + 14·45-s − 18·47-s + 13·49-s + 3·51-s + 5·53-s + 20·55-s − 6·57-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.894·5-s − 1.51·7-s + 7/3·9-s + 3.01·11-s − 1.38·13-s − 1.54·15-s − 0.242·17-s + 0.458·19-s + 2.61·21-s + 2.29·23-s + 6/5·25-s − 3.46·27-s − 1.11·29-s − 5.22·33-s − 1.35·35-s + 0.493·37-s + 2.40·39-s + 3.12·41-s + 3.96·43-s + 2.08·45-s − 2.62·47-s + 13/7·49-s + 0.420·51-s + 0.686·53-s + 2.69·55-s − 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 43^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 43^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.642999258\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.642999258\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| good | 3 | $D_4\times C_2$ | \( 1 + p T + 2 T^{2} + p T^{3} + 13 T^{4} + p^{2} T^{5} + 2 p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 - 2 T - 2 T^{2} + 8 T^{3} - 9 T^{4} + 8 p T^{5} - 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 4 T + 3 T^{2} - 4 T^{3} + 8 T^{4} - 4 p T^{5} + 3 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 5 T + 27 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 5 T - 6 T^{2} + 25 T^{3} + 467 T^{4} + 25 p T^{5} - 6 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + T - 2 T^{2} - 31 T^{3} - 297 T^{4} - 31 p T^{5} - 2 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 2 T - 30 T^{2} + 8 T^{3} + 719 T^{4} + 8 p T^{5} - 30 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 11 T + 2 p T^{2} - 319 T^{3} + 2313 T^{4} - 319 p T^{5} + 2 p^{3} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 3 T - 56 T^{2} + 27 T^{3} + 2523 T^{4} + 27 p T^{5} - 56 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 10 T + 87 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 + 9 T + 103 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 5 T - 86 T^{2} - 25 T^{3} + 8187 T^{4} - 25 p T^{5} - 86 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - T + 87 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - T - 110 T^{2} + 11 T^{3} + 8539 T^{4} + 11 p T^{5} - 110 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + T - 122 T^{2} - 11 T^{3} + 10573 T^{4} - 11 p T^{5} - 122 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 3 T + 16 T^{2} + 447 T^{3} - 5631 T^{4} + 447 p T^{5} + 16 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 3 T - 128 T^{2} + 27 T^{3} + 12783 T^{4} + 27 p T^{5} - 128 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 5 T - 138 T^{2} + 25 T^{3} + 18353 T^{4} + 25 p T^{5} - 138 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 3 T + 52 T^{2} + 627 T^{3} - 5787 T^{4} + 627 p T^{5} + 52 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 3 T - 160 T^{2} - 27 T^{3} + 19839 T^{4} - 27 p T^{5} - 160 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 14 T + 238 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.49937664576005641230588482778, −7.01813724930605880428625763430, −6.82682395476057793983019899233, −6.81515685889756900098772470019, −6.72049559043565511753604906659, −6.28381117976259863957200415080, −6.28139288400936353671126413303, −5.91070079527351790317020071581, −5.56505670930235103691182477619, −5.47106484423715376105688397300, −5.28389221529602260715502623056, −5.08136394797352630081162618916, −4.57575446471385514120899861455, −4.11622016275687595639130547902, −4.07103866708086323633214627811, −4.05807445298138916146144196047, −3.84481178045795871696941462992, −3.07283095897408718613083925455, −2.86527865341171090174916327002, −2.52375011504096944974253807479, −2.40833198999483307811145547683, −1.49855150716976584606688419146, −1.30822638810636183787085016795, −1.10196448655760376159716388555, −0.47272693966423748827090238065,
0.47272693966423748827090238065, 1.10196448655760376159716388555, 1.30822638810636183787085016795, 1.49855150716976584606688419146, 2.40833198999483307811145547683, 2.52375011504096944974253807479, 2.86527865341171090174916327002, 3.07283095897408718613083925455, 3.84481178045795871696941462992, 4.05807445298138916146144196047, 4.07103866708086323633214627811, 4.11622016275687595639130547902, 4.57575446471385514120899861455, 5.08136394797352630081162618916, 5.28389221529602260715502623056, 5.47106484423715376105688397300, 5.56505670930235103691182477619, 5.91070079527351790317020071581, 6.28139288400936353671126413303, 6.28381117976259863957200415080, 6.72049559043565511753604906659, 6.81515685889756900098772470019, 6.82682395476057793983019899233, 7.01813724930605880428625763430, 7.49937664576005641230588482778
