| L(s) = 1 | + 2-s − 2·3-s − 3·4-s − 5-s − 2·6-s + 7-s − 5·8-s + 3·9-s − 10-s + 6·12-s − 4·13-s + 14-s + 2·15-s + 10·17-s + 3·18-s − 9·19-s + 3·20-s − 2·21-s − 2·23-s + 10·24-s + 5·25-s − 4·26-s − 10·27-s − 3·28-s − 29-s + 2·30-s + 7·31-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s − 3/2·4-s − 0.447·5-s − 0.816·6-s + 0.377·7-s − 1.76·8-s + 9-s − 0.316·10-s + 1.73·12-s − 1.10·13-s + 0.267·14-s + 0.516·15-s + 2.42·17-s + 0.707·18-s − 2.06·19-s + 0.670·20-s − 0.436·21-s − 0.417·23-s + 2.04·24-s + 25-s − 0.784·26-s − 1.92·27-s − 0.566·28-s − 0.185·29-s + 0.365·30-s + 1.25·31-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
| \(L(1)\) |
\(\approx\) |
\(1.333240516\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.333240516\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_4\times C_2$ | \( 1 - T + p^{2} T^{2} - p T^{3} + 9 T^{4} - p^{2} T^{5} + p^{4} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 3 | $C_2^2:C_4$ | \( 1 + 2 T + T^{2} + 2 p T^{3} + 19 T^{4} + 2 p^{2} T^{5} + p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 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 | $C_2^2:C_4$ | \( 1 + 4 T + 3 T^{2} + 50 T^{3} + 341 T^{4} + 50 p T^{5} + 3 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 - 10 T + 83 T^{2} - 460 T^{3} + 2189 T^{4} - 460 p T^{5} + 83 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 + 9 T + 12 T^{2} - 163 T^{3} - 1095 T^{4} - 163 p T^{5} + 12 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + T + 15 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2:C_4$ | \( 1 + T - 13 T^{2} - 137 T^{3} + 440 T^{4} - 137 p T^{5} - 13 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 - 7 T - 12 T^{2} + 121 T^{3} + 125 T^{4} + 121 p T^{5} - 12 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 - 8 T + 27 T^{2} - 340 T^{3} + 3401 T^{4} - 340 p T^{5} + 27 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_4\times C_2$ | \( 1 - 25 T + 334 T^{2} - 3125 T^{3} + 22431 T^{4} - 3125 p T^{5} + 334 p^{2} T^{6} - 25 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 5 T - 9 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 - 3 T - 13 T^{2} + 285 T^{3} + 136 T^{4} + 285 p T^{5} - 13 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 - T - 37 T^{2} + 305 T^{3} + 1976 T^{4} + 305 p T^{5} - 37 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 - 7 T + 65 T^{2} - 667 T^{3} + 8084 T^{4} - 667 p T^{5} + 65 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 + 9 T - 15 T^{2} - 629 T^{3} - 4236 T^{4} - 629 p T^{5} - 15 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + T + 73 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2:C_4$ | \( 1 - 23 T + 133 T^{2} + 1649 T^{3} - 28620 T^{4} + 1649 p T^{5} + 133 p^{2} T^{6} - 23 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 + 7 T + 86 T^{2} + 611 T^{3} + 2239 T^{4} + 611 p T^{5} + 86 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2:C_4$ | \( 1 - 5 T - 19 T^{2} + 635 T^{3} - 1004 T^{4} + 635 p T^{5} - 19 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 - 11 T + 68 T^{2} - 1215 T^{3} + 17441 T^{4} - 1215 p T^{5} + 68 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 7 T + 179 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_4\times C_2$ | \( 1 + 7 T - 48 T^{2} - 1015 T^{3} - 2449 T^{4} - 1015 p T^{5} - 48 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.42389649050360230054976398159, −6.89251889844126417113709533681, −6.75200150617430575704652226130, −6.72329936027291924424179686576, −6.22852980988965794969949852094, −5.98756815216516044487358143291, −5.82911030328767391988339822624, −5.62429003646134896470202971182, −5.55323176307026255024625977294, −5.13490570272589063755131981888, −4.96032437719084963395963563240, −4.63057335794370586731683015635, −4.38376659553352797589243796200, −4.33816420339235268186990215272, −4.10501542245901849198901023347, −3.84589515454991701089922081342, −3.69410900570686441518810688038, −3.07255999996015060291059839037, −2.85634687171925694623744405559, −2.40077699521415150554766582603, −2.39395859375520437544504714609, −1.76724092426401294873955518281, −1.07098023683327170351669384081, −0.72078681363540739656213352393, −0.47461562048298757790386180662,
0.47461562048298757790386180662, 0.72078681363540739656213352393, 1.07098023683327170351669384081, 1.76724092426401294873955518281, 2.39395859375520437544504714609, 2.40077699521415150554766582603, 2.85634687171925694623744405559, 3.07255999996015060291059839037, 3.69410900570686441518810688038, 3.84589515454991701089922081342, 4.10501542245901849198901023347, 4.33816420339235268186990215272, 4.38376659553352797589243796200, 4.63057335794370586731683015635, 4.96032437719084963395963563240, 5.13490570272589063755131981888, 5.55323176307026255024625977294, 5.62429003646134896470202971182, 5.82911030328767391988339822624, 5.98756815216516044487358143291, 6.22852980988965794969949852094, 6.72329936027291924424179686576, 6.75200150617430575704652226130, 6.89251889844126417113709533681, 7.42389649050360230054976398159
