| L(s) = 1 | + 2·4-s − 5-s + 7-s − 2·9-s − 9·13-s − 12·17-s − 13·19-s − 2·20-s − 4·23-s + 5·25-s + 2·28-s − 7·29-s − 5·31-s − 35-s − 4·36-s + 17·37-s − 11·41-s − 43-s + 2·45-s + 6·47-s − 18·52-s − 12·53-s − 16·59-s − 8·61-s − 2·63-s + 9·65-s + 8·67-s + ⋯ |
| L(s) = 1 | + 4-s − 0.447·5-s + 0.377·7-s − 2/3·9-s − 2.49·13-s − 2.91·17-s − 2.98·19-s − 0.447·20-s − 0.834·23-s + 25-s + 0.377·28-s − 1.29·29-s − 0.898·31-s − 0.169·35-s − 2/3·36-s + 2.79·37-s − 1.71·41-s − 0.152·43-s + 0.298·45-s + 0.875·47-s − 2.49·52-s − 1.64·53-s − 2.08·59-s − 1.02·61-s − 0.251·63-s + 1.11·65-s + 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{4} \cdot 41^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{4} \cdot 41^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.03468192608\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03468192608\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 41 | $C_4$ | \( 1 + 11 T + 111 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| good | 2 | $C_4\times C_2$ | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 3 | $C_2^2$ | \( ( 1 + T^{2} + p^{2} T^{4} )^{2} \) |
| 5 | $C_4\times C_2$ | \( 1 + T - 4 T^{2} - 9 T^{3} + 11 T^{4} - 9 p T^{5} - 4 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^2:C_4$ | \( 1 - T^{2} + 30 T^{3} + 91 T^{4} + 30 p T^{5} - p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 + 9 T + 18 T^{2} - 115 T^{3} - 789 T^{4} - 115 p T^{5} + 18 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 + 12 T + 77 T^{2} + 420 T^{3} + 1981 T^{4} + 420 p T^{5} + 77 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 + 13 T + 50 T^{2} - 37 T^{3} - 791 T^{4} - 37 p T^{5} + 50 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2:C_4$ | \( 1 + 4 T + p T^{2} + 150 T^{3} + 1151 T^{4} + 150 p T^{5} + p^{3} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2:C_4$ | \( 1 + 7 T + 5 T^{2} + 117 T^{3} + 1484 T^{4} + 117 p T^{5} + 5 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 + 5 T + 9 T^{2} + 205 T^{3} + 1916 T^{4} + 205 p T^{5} + 9 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 - 17 T + 72 T^{2} + 605 T^{3} - 7669 T^{4} + 605 p T^{5} + 72 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2^2:C_4$ | \( 1 + T + 8 T^{2} - 175 T^{3} + 961 T^{4} - 175 p T^{5} + 8 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2^2:C_4$ | \( 1 - 6 T - 31 T^{2} + 318 T^{3} - 161 T^{4} + 318 p T^{5} - 31 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 + 12 T + 11 T^{2} - 84 T^{3} + 1369 T^{4} - 84 p T^{5} + 11 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 + 16 T + 47 T^{2} - 942 T^{3} - 12445 T^{4} - 942 p T^{5} + 47 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 + 8 T - 27 T^{2} + 16 T^{3} + 4025 T^{4} + 16 p T^{5} - 27 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_4\times C_2$ | \( 1 - 8 T - 3 T^{2} + 560 T^{3} - 4279 T^{4} + 560 p T^{5} - 3 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_4\times C_2$ | \( 1 + 22 T + 313 T^{2} + 3674 T^{3} + 36105 T^{4} + 3674 p T^{5} + 313 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 11 T + 115 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 8 T + 169 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{4} \) |
| 89 | $C_2^2:C_4$ | \( 1 + 29 T + 367 T^{2} + 3607 T^{3} + 35640 T^{4} + 3607 p T^{5} + 367 p^{2} T^{6} + 29 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^2:C_4$ | \( 1 + T + 44 T^{2} + 587 T^{3} + 8959 T^{4} + 587 p T^{5} + 44 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.679966304207036966162357686327, −8.347862031695501256352547161054, −8.090288058065565087465671664152, −7.925754324728750825277420796620, −7.38992823983176181300293567570, −7.25555153212601329551119596861, −7.01711488393384363519088720490, −7.01560060896927248906610564447, −6.48597084713485371484289146870, −6.11620771461763941210175105239, −6.08561181962945822486297601784, −6.03140978654089467330493000953, −5.24265414906733168564543342404, −5.05763581699274236598609621167, −4.60628803576230124066531717286, −4.38719675407319583878780639613, −4.36810882369573082562899974044, −4.11930577537592518722367611845, −3.31755718748994953505151448806, −2.97586025273578933713172011842, −2.67855151998531586328515079829, −2.05775835243454030106773562796, −2.02227895550259129114930214339, −1.99974282907645260136250969811, −0.07624734508162267726931561773,
0.07624734508162267726931561773, 1.99974282907645260136250969811, 2.02227895550259129114930214339, 2.05775835243454030106773562796, 2.67855151998531586328515079829, 2.97586025273578933713172011842, 3.31755718748994953505151448806, 4.11930577537592518722367611845, 4.36810882369573082562899974044, 4.38719675407319583878780639613, 4.60628803576230124066531717286, 5.05763581699274236598609621167, 5.24265414906733168564543342404, 6.03140978654089467330493000953, 6.08561181962945822486297601784, 6.11620771461763941210175105239, 6.48597084713485371484289146870, 7.01560060896927248906610564447, 7.01711488393384363519088720490, 7.25555153212601329551119596861, 7.38992823983176181300293567570, 7.925754324728750825277420796620, 8.090288058065565087465671664152, 8.347862031695501256352547161054, 8.679966304207036966162357686327
