| L(s) = 1 | − 8·4-s + 5·7-s + 5·13-s + 40·16-s + 19-s − 40·28-s + 7·31-s − 10·37-s + 5·43-s + 7·49-s − 40·52-s + 13·61-s − 160·64-s + 5·67-s − 10·73-s − 8·76-s + 4·79-s + 25·91-s + 5·97-s + 20·103-s + 19·109-s + 200·112-s − 44·121-s − 56·124-s + 127-s + 131-s + 5·133-s + ⋯ |
| L(s) = 1 | − 4·4-s + 1.88·7-s + 1.38·13-s + 10·16-s + 0.229·19-s − 7.55·28-s + 1.25·31-s − 1.64·37-s + 0.762·43-s + 49-s − 5.54·52-s + 1.66·61-s − 20·64-s + 0.610·67-s − 1.17·73-s − 0.917·76-s + 0.450·79-s + 2.62·91-s + 0.507·97-s + 1.97·103-s + 1.81·109-s + 18.8·112-s − 4·121-s − 5.02·124-s + 0.0887·127-s + 0.0873·131-s + 0.433·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.837633398\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.837633398\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 7 | $C_4\times C_2$ | \( 1 - 5 T + 18 T^{2} - 55 T^{3} + 149 T^{4} - 55 p T^{5} + 18 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 13 | $C_4\times C_2$ | \( 1 - 5 T + 12 T^{2} + 5 T^{3} - 181 T^{4} + 5 p T^{5} + 12 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 19 | $C_4\times C_2$ | \( 1 - T - 18 T^{2} + 37 T^{3} + 305 T^{4} + 37 p T^{5} - 18 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_4\times C_2$ | \( 1 - 7 T + 18 T^{2} + 91 T^{3} - 1195 T^{4} + 91 p T^{5} + 18 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_4\times C_2$ | \( 1 + 10 T + 63 T^{2} + 260 T^{3} + 269 T^{4} + 260 p T^{5} + 63 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_4\times C_2$ | \( 1 - 5 T - 18 T^{2} + 305 T^{3} - 751 T^{4} + 305 p T^{5} - 18 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_4\times C_2$ | \( 1 - 13 T + 108 T^{2} - 611 T^{3} + 1355 T^{4} - 611 p T^{5} + 108 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_4\times C_2$ | \( 1 - 5 T - 42 T^{2} + 545 T^{3} + 89 T^{4} + 545 p T^{5} - 42 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 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 | $C_4\times C_2$ | \( 1 - 4 T - 63 T^{2} + 568 T^{3} + 2705 T^{4} + 568 p T^{5} - 63 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 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
−5.70104144590741387380304735079, −5.37724994270796124755700667475, −5.25328213677098490934792860784, −5.18308026795681179203300052676, −4.96166231440275491666564431620, −4.59276872608395175820103553342, −4.52395776531428781518635742644, −4.52040898120177382248774536471, −4.31261814212534735791202633302, −4.07844235176578145474635722504, −3.78919367653310266980153369383, −3.74569790852699295623225361722, −3.58310715666819786980000485565, −3.17277116050041897538289588722, −3.03456504837728613466528575081, −2.97001974974080423295538867395, −2.61633833591290146210116364077, −1.89936695853322530431784589468, −1.81975447709519974828361640789, −1.59547749651965825358137188228, −1.59369308106413737614869181265, −0.851009695732870467375459054913, −0.77632816047039006677937070142, −0.67043959111145822753039871658, −0.43433093463053435861365849590,
0.43433093463053435861365849590, 0.67043959111145822753039871658, 0.77632816047039006677937070142, 0.851009695732870467375459054913, 1.59369308106413737614869181265, 1.59547749651965825358137188228, 1.81975447709519974828361640789, 1.89936695853322530431784589468, 2.61633833591290146210116364077, 2.97001974974080423295538867395, 3.03456504837728613466528575081, 3.17277116050041897538289588722, 3.58310715666819786980000485565, 3.74569790852699295623225361722, 3.78919367653310266980153369383, 4.07844235176578145474635722504, 4.31261814212534735791202633302, 4.52040898120177382248774536471, 4.52395776531428781518635742644, 4.59276872608395175820103553342, 4.96166231440275491666564431620, 5.18308026795681179203300052676, 5.25328213677098490934792860784, 5.37724994270796124755700667475, 5.70104144590741387380304735079
