| L(s) = 1 | + 4·3-s + 8·7-s + 8·9-s + 8·13-s + 8·17-s + 20·19-s + 32·21-s − 8·23-s + 8·27-s − 24·37-s + 32·39-s + 20·41-s − 8·43-s + 32·49-s + 32·51-s − 16·53-s + 80·57-s + 8·59-s − 16·61-s + 64·63-s + 12·67-s − 32·69-s − 8·73-s + 16·79-s − 7·81-s + 4·83-s + 64·91-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 3.02·7-s + 8/3·9-s + 2.21·13-s + 1.94·17-s + 4.58·19-s + 6.98·21-s − 1.66·23-s + 1.53·27-s − 3.94·37-s + 5.12·39-s + 3.12·41-s − 1.21·43-s + 32/7·49-s + 4.48·51-s − 2.19·53-s + 10.5·57-s + 1.04·59-s − 2.04·61-s + 8.06·63-s + 1.46·67-s − 3.85·69-s − 0.936·73-s + 1.80·79-s − 7/9·81-s + 0.439·83-s + 6.70·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(29.81187165\) |
| \(L(\frac12)\) |
\(\approx\) |
\(29.81187165\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^3$ | \( 1 - 4 T + 8 T^{2} - 8 T^{3} + 7 T^{4} - 8 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 30 T^{2} + 443 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 72 T^{3} + 146 T^{4} - 72 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 176 T^{3} + 943 T^{4} - 176 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 10 T + 3 p T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 152 T^{3} + 706 T^{4} + 152 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 60 T^{2} + 2198 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 2694 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{4} \) |
| 43 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 24 T^{3} - 1582 T^{4} + 24 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2^3$ | \( 1 - 4222 T^{4} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 1168 T^{3} + 10258 T^{4} + 1168 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 4 T + 26 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 8 T + 114 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 984 T^{3} + 13223 T^{4} - 984 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 138 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 432 T^{3} + 5471 T^{4} + 432 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 8 T + 150 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 232 T^{3} + 6103 T^{4} - 232 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 210 T^{2} + 22163 T^{4} - 210 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^3$ | \( 1 + 2498 T^{4} + p^{4} T^{8} \) |
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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
−6.98221568294987898347919165209, −6.46125310710533245818852460470, −6.09573111746328496802852182257, −6.03476688401065034078046344679, −5.96947992830076153152988994684, −5.27621340114071343997005403072, −5.24793839272349673776392682753, −5.19280261854960819453798811774, −5.18191472582528517818001868250, −4.90867725742183946560705828684, −4.23519811919256876401453829141, −4.10637224423941451047973956366, −4.07474417331806043410335434127, −3.49644767181320993648663528174, −3.49014803788208128925230710622, −3.43645767978703212484514309136, −3.04933614909202244520993750939, −2.88759356800903112260310520184, −2.52245838114601384460427666238, −1.95543136562099562637985653098, −1.90537748275769706804373629975, −1.48390110493691424403311188897, −1.32077755789816994286839433995, −1.27457257402089551913919111158, −0.78609668938360413940362592892,
0.78609668938360413940362592892, 1.27457257402089551913919111158, 1.32077755789816994286839433995, 1.48390110493691424403311188897, 1.90537748275769706804373629975, 1.95543136562099562637985653098, 2.52245838114601384460427666238, 2.88759356800903112260310520184, 3.04933614909202244520993750939, 3.43645767978703212484514309136, 3.49014803788208128925230710622, 3.49644767181320993648663528174, 4.07474417331806043410335434127, 4.10637224423941451047973956366, 4.23519811919256876401453829141, 4.90867725742183946560705828684, 5.18191472582528517818001868250, 5.19280261854960819453798811774, 5.24793839272349673776392682753, 5.27621340114071343997005403072, 5.96947992830076153152988994684, 6.03476688401065034078046344679, 6.09573111746328496802852182257, 6.46125310710533245818852460470, 6.98221568294987898347919165209
