| L(s) = 1 | − 4·7-s + 2·9-s − 4·13-s − 4·17-s + 12·23-s − 8·29-s + 12·37-s + 4·43-s + 20·47-s + 8·49-s + 12·53-s + 16·59-s + 24·61-s − 8·63-s + 12·67-s − 4·73-s − 5·81-s − 4·83-s + 40·89-s + 16·91-s − 4·97-s − 28·103-s − 44·107-s − 4·113-s − 8·117-s + 16·119-s + 20·121-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 2/3·9-s − 1.10·13-s − 0.970·17-s + 2.50·23-s − 1.48·29-s + 1.97·37-s + 0.609·43-s + 2.91·47-s + 8/7·49-s + 1.64·53-s + 2.08·59-s + 3.07·61-s − 1.00·63-s + 1.46·67-s − 0.468·73-s − 5/9·81-s − 0.439·83-s + 4.23·89-s + 1.67·91-s − 0.406·97-s − 2.75·103-s − 4.25·107-s − 0.376·113-s − 0.739·117-s + 1.46·119-s + 1.81·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.121150337\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.121150337\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 20 T^{3} + 46 T^{4} + 20 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 214 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} - 4 T^{3} - 194 T^{4} - 4 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 12 T^{3} - 178 T^{4} + 12 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 1270 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 444 T^{3} + 2542 T^{4} - 444 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 468 T^{3} + 3038 T^{4} - 468 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 164 T^{3} + 3358 T^{4} - 164 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 20 T + 200 T^{2} - 1860 T^{3} + 15182 T^{4} - 1860 p T^{5} + 200 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 61 | $D_{4}$ | \( ( 1 - 12 T + 126 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} + 180 T^{3} - 6274 T^{4} + 180 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 132 T^{2} + 13286 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 44 T^{3} - 3602 T^{4} + 44 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 292 T^{2} + 33670 T^{4} - 292 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 196 T^{3} + 3646 T^{4} + 196 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 20 T + 246 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.00445985598341461780265280614, −6.90253930288559556773994584391, −6.82829773912297807694231438333, −6.29642219959939020306772271874, −6.16206731111710385031210453306, −5.60373077247507310356585937024, −5.57817122955142603615790371469, −5.50741524443740044845264631277, −5.48136026305845552081475456766, −4.76281350849780962409744014883, −4.72840169249165702237381466792, −4.47602736197199566759139612991, −4.06637819543971443540780013244, −4.04304557074583223127565870918, −3.73746855740865562908603316494, −3.46248613724721306978545722969, −3.23169825624084040071190781216, −2.68414846342480031005116809245, −2.60092252171982828914386518938, −2.37532715194180432980707942282, −2.22636124926232743758354934975, −1.74592835914342932007214026742, −0.959123601835265283717060823176, −0.835671766477589081285097403245, −0.51891344622334452814825754826,
0.51891344622334452814825754826, 0.835671766477589081285097403245, 0.959123601835265283717060823176, 1.74592835914342932007214026742, 2.22636124926232743758354934975, 2.37532715194180432980707942282, 2.60092252171982828914386518938, 2.68414846342480031005116809245, 3.23169825624084040071190781216, 3.46248613724721306978545722969, 3.73746855740865562908603316494, 4.04304557074583223127565870918, 4.06637819543971443540780013244, 4.47602736197199566759139612991, 4.72840169249165702237381466792, 4.76281350849780962409744014883, 5.48136026305845552081475456766, 5.50741524443740044845264631277, 5.57817122955142603615790371469, 5.60373077247507310356585937024, 6.16206731111710385031210453306, 6.29642219959939020306772271874, 6.82829773912297807694231438333, 6.90253930288559556773994584391, 7.00445985598341461780265280614
