| L(s) = 1 | − 2·2-s − 10·3-s + 6·4-s − 10·5-s + 20·6-s + 4·8-s + 79·9-s + 20·10-s − 66·11-s − 60·12-s + 20·13-s + 100·15-s + 24·16-s − 70·17-s − 158·18-s − 140·19-s − 60·20-s + 132·22-s + 16·23-s − 40·24-s + 25·25-s − 40·26-s − 830·27-s − 516·29-s − 200·30-s + 20·31-s + 120·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.92·3-s + 3/4·4-s − 0.894·5-s + 1.36·6-s + 0.176·8-s + 2.92·9-s + 0.632·10-s − 1.80·11-s − 1.44·12-s + 0.426·13-s + 1.72·15-s + 3/8·16-s − 0.998·17-s − 2.06·18-s − 1.69·19-s − 0.670·20-s + 1.27·22-s + 0.145·23-s − 0.340·24-s + 1/5·25-s − 0.301·26-s − 5.91·27-s − 3.30·29-s − 1.21·30-s + 0.115·31-s + 0.662·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2107379670\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2107379670\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 + p T - p T^{2} - 5 p^{2} T^{3} - 15 p^{2} T^{4} - 5 p^{5} T^{5} - p^{7} T^{6} + p^{10} T^{7} + p^{12} T^{8} \) |
| 3 | $C_2^2$ | \( ( 1 + 5 T - 2 T^{2} + 5 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 + 6 p T + 71 p T^{2} + 498 p^{2} T^{3} + 43068 p^{2} T^{4} + 498 p^{5} T^{5} + 71 p^{7} T^{6} + 6 p^{10} T^{7} + p^{12} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 10 T + 19 T^{2} - 10 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 70 T - 103 p T^{2} - 222250 T^{3} - 3975468 T^{4} - 222250 p^{3} T^{5} - 103 p^{7} T^{6} + 70 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 140 T + 5382 T^{2} + 70000 T^{3} + 20669243 T^{4} + 70000 p^{3} T^{5} + 5382 p^{6} T^{6} + 140 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 16 T - 15518 T^{2} + 136960 T^{3} + 97668435 T^{4} + 136960 p^{3} T^{5} - 15518 p^{6} T^{6} - 16 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 258 T + 40075 T^{2} + 258 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 20 T - 19682 T^{2} + 790000 T^{3} - 496133357 T^{4} + 790000 p^{3} T^{5} - 19682 p^{6} T^{6} - 20 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 328 T + 678 T^{2} + 1836800 T^{3} + 3413714075 T^{4} + 1836800 p^{3} T^{5} + 678 p^{6} T^{6} + 328 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 300 T + 50342 T^{2} - 300 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 116 T + 160794 T^{2} + 116 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 30 T - 189371 T^{2} + 521250 T^{3} + 25330397412 T^{4} + 521250 p^{3} T^{5} - 189371 p^{6} T^{6} - 30 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 540 T + 79346 T^{2} - 46170000 T^{3} - 20525133813 T^{4} - 46170000 p^{3} T^{5} + 79346 p^{6} T^{6} + 540 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 380 T - 284858 T^{2} + 7030000 T^{3} + 112425169323 T^{4} + 7030000 p^{3} T^{5} - 284858 p^{6} T^{6} + 380 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 1080 T + 460438 T^{2} + 272160000 T^{3} + 182111332683 T^{4} + 272160000 p^{3} T^{5} + 460438 p^{6} T^{6} + 1080 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 468 T - 335882 T^{2} - 21818160 T^{3} + 151587971355 T^{4} - 21818160 p^{3} T^{5} - 335882 p^{6} T^{6} + 468 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 1056 T + 949550 T^{2} + 1056 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 860 T + 216666 T^{2} + 219386000 T^{3} - 165591231133 T^{4} + 219386000 p^{3} T^{5} + 216666 p^{6} T^{6} - 860 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 2 p T + 364789 T^{2} - 2651806 p T^{3} - 139914650492 T^{4} - 2651806 p^{4} T^{5} + 364789 p^{6} T^{6} + 2 p^{10} T^{7} + p^{12} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 40 T + 703974 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 240 T + 221662 T^{2} + 377760000 T^{3} - 510671165517 T^{4} + 377760000 p^{3} T^{5} + 221662 p^{6} T^{6} - 240 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 1630 T + 2133171 T^{2} - 1630 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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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
−8.139236716743838736340029136617, −7.945012703827366374742883711340, −7.73987482626082993943517757335, −7.53630507714398640574808927602, −7.30092563482984294736755038543, −7.08880741765671910998399862954, −7.02249871735022836574604880108, −6.32338670133726264128085749725, −6.07431957106300509556175510381, −5.84521017209001430072368745616, −5.79674914756520969338450279907, −5.64113954055295339627182907295, −4.79333828501458035574750507644, −4.70394493366525851689294643254, −4.51772387049183169469471349227, −4.38935713784734530194269727193, −3.62310471615146745117622319966, −3.48726832547136692592757227362, −3.25313386057147293321998409899, −2.18898454662478548698924394158, −2.04219889507036837077853262329, −1.85684755041205269703743474758, −1.27983836096017389616939558613, −0.37261880346719232517167380483, −0.25958062274990615940494059506,
0.25958062274990615940494059506, 0.37261880346719232517167380483, 1.27983836096017389616939558613, 1.85684755041205269703743474758, 2.04219889507036837077853262329, 2.18898454662478548698924394158, 3.25313386057147293321998409899, 3.48726832547136692592757227362, 3.62310471615146745117622319966, 4.38935713784734530194269727193, 4.51772387049183169469471349227, 4.70394493366525851689294643254, 4.79333828501458035574750507644, 5.64113954055295339627182907295, 5.79674914756520969338450279907, 5.84521017209001430072368745616, 6.07431957106300509556175510381, 6.32338670133726264128085749725, 7.02249871735022836574604880108, 7.08880741765671910998399862954, 7.30092563482984294736755038543, 7.53630507714398640574808927602, 7.73987482626082993943517757335, 7.945012703827366374742883711340, 8.139236716743838736340029136617
