| L(s) = 1 | − 2·3-s − 3·4-s − 6·5-s + 10·7-s − 9-s − 8·11-s + 6·12-s + 12·15-s + 4·16-s + 12·17-s + 16·19-s + 18·20-s − 20·21-s − 8·23-s + 11·25-s + 2·27-s − 30·28-s + 20·29-s + 2·31-s + 16·33-s − 60·35-s + 3·36-s − 8·37-s + 16·41-s + 24·44-s + 6·45-s − 10·47-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 3/2·4-s − 2.68·5-s + 3.77·7-s − 1/3·9-s − 2.41·11-s + 1.73·12-s + 3.09·15-s + 16-s + 2.91·17-s + 3.67·19-s + 4.02·20-s − 4.36·21-s − 1.66·23-s + 11/5·25-s + 0.384·27-s − 5.66·28-s + 3.71·29-s + 0.359·31-s + 2.78·33-s − 10.1·35-s + 1/2·36-s − 1.31·37-s + 2.49·41-s + 3.61·44-s + 0.894·45-s − 1.45·47-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
| \(L(1)\) |
\(\approx\) |
\(0.7116160720\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7116160720\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| good | 2 | $C_2^3$ | \( 1 + 3 T^{2} + 5 T^{4} + 3 p^{2} T^{6} + p^{4} T^{8} \) |
| 3 | $D_4\times C_2$ | \( 1 + 2 T + 5 T^{2} + 10 T^{3} + 16 T^{4} + 10 p T^{5} + 5 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2^2$ | \( ( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 + 8 T + 41 T^{2} + 152 T^{3} + 532 T^{4} + 152 p T^{5} + 41 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^3$ | \( 1 + 62 T^{4} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 12 T + 36 T^{2} + 12 p T^{3} - 1777 T^{4} + 12 p^{2} T^{5} + 36 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2$$\times$$C_2^2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 11 T^{2} + p^{2} T^{4} ) \) |
| 23 | $D_4\times C_2$ | \( 1 + 8 T + 14 T^{2} + 32 T^{3} + 499 T^{4} + 32 p T^{5} + 14 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 20 T + 200 T^{2} - 1460 T^{3} + 8722 T^{4} - 1460 p T^{5} + 200 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 2 T - 56 T^{2} + 4 T^{3} + 2515 T^{4} + 4 p T^{5} - 56 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 8 T - 14 T^{2} + 32 T^{3} + 2347 T^{4} + 32 p T^{5} - 14 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 8406 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 10 T + 74 T^{2} + 496 T^{3} + 2599 T^{4} + 496 p T^{5} + 74 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 2 T + 2 T^{2} - 208 T^{3} - 3017 T^{4} - 208 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 18 T + 128 T^{2} + 1404 T^{3} + 15819 T^{4} + 1404 p T^{5} + 128 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^3$ | \( 1 + 73 T^{2} + 1608 T^{4} + 73 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2^3$ | \( 1 - 6 T + 18 T^{2} - 36 T^{3} - 3649 T^{4} - 36 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 30 T + 450 T^{2} - 5460 T^{3} + 53927 T^{4} - 5460 p T^{5} + 450 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 12 T + 142 T^{2} - 1128 T^{3} + 7011 T^{4} - 1128 p T^{5} + 142 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 24 T + 225 T^{2} - 768 T^{3} - 532 T^{4} - 768 p T^{5} + 225 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 2 T + 164 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 + 10 T + 50 T^{2} - 1280 T^{3} - 14321 T^{4} - 1280 p T^{5} + 50 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 648 T^{3} + 12866 T^{4} + 648 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.326900051272817066089826906977, −8.105951383870364837114297515224, −8.067554585563199438255799587719, −7.83581135101102614985336308868, −7.79969975409364441022558859394, −7.67899981794446296763254117928, −7.41194891929719928575938173058, −7.09818640262132342470421212234, −6.42136846021177150235846990349, −5.97889491626366333250614007184, −5.56108369038254359687301009321, −5.44704789319198853249801868324, −5.38147013761258229882730023092, −4.90857876017896760883263956844, −4.89562546218098390002880026239, −4.68916694910153168880059551447, −4.44032515007122986701165539456, −3.84005659900998156308865243467, −3.63655365298382858686317214009, −3.36099102996729294592844219173, −2.86680015480360249496612369712, −2.43461356258306072715760563754, −1.55642298247629892690490389631, −0.830326953249869851937098281622, −0.73395917718888594622106665778,
0.73395917718888594622106665778, 0.830326953249869851937098281622, 1.55642298247629892690490389631, 2.43461356258306072715760563754, 2.86680015480360249496612369712, 3.36099102996729294592844219173, 3.63655365298382858686317214009, 3.84005659900998156308865243467, 4.44032515007122986701165539456, 4.68916694910153168880059551447, 4.89562546218098390002880026239, 4.90857876017896760883263956844, 5.38147013761258229882730023092, 5.44704789319198853249801868324, 5.56108369038254359687301009321, 5.97889491626366333250614007184, 6.42136846021177150235846990349, 7.09818640262132342470421212234, 7.41194891929719928575938173058, 7.67899981794446296763254117928, 7.79969975409364441022558859394, 7.83581135101102614985336308868, 8.067554585563199438255799587719, 8.105951383870364837114297515224, 8.326900051272817066089826906977
