| L(s) = 1 | − 16·7-s + 16·11-s − 12·13-s − 44·17-s − 16·23-s + 80·31-s + 108·37-s + 32·41-s + 48·43-s + 96·47-s + 128·49-s − 100·53-s − 96·61-s − 16·67-s − 64·71-s + 188·73-s − 256·77-s − 9·81-s + 192·83-s + 192·91-s − 132·97-s − 200·101-s − 32·103-s + 80·107-s − 172·113-s + 704·119-s − 132·121-s + ⋯ |
| L(s) = 1 | − 2.28·7-s + 1.45·11-s − 0.923·13-s − 2.58·17-s − 0.695·23-s + 2.58·31-s + 2.91·37-s + 0.780·41-s + 1.11·43-s + 2.04·47-s + 2.61·49-s − 1.88·53-s − 1.57·61-s − 0.238·67-s − 0.901·71-s + 2.57·73-s − 3.32·77-s − 1/9·81-s + 2.31·83-s + 2.10·91-s − 1.36·97-s − 1.98·101-s − 0.310·103-s + 0.747·107-s − 1.52·113-s + 5.91·119-s − 1.09·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9731895697\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9731895697\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 528 T^{3} + 1922 T^{4} + 528 p^{2} T^{5} + 128 p^{4} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 8 T + 162 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} - 60 T^{3} - 30226 T^{4} - 60 p^{2} T^{5} + 72 p^{4} T^{6} + 12 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 44 T + 968 T^{2} + 21252 T^{3} + 428942 T^{4} + 21252 p^{2} T^{5} + 968 p^{4} T^{6} + 44 p^{6} T^{7} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 740 T^{2} + 299238 T^{4} - 740 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 2064 T^{3} - 126718 T^{4} + 2064 p^{2} T^{5} + 128 p^{4} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 2660 T^{2} + 3085158 T^{4} - 2660 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 40 T + 1938 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 54 T + 1458 T^{2} - 54 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 16 T + 3330 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 48 T + 1152 T^{2} - 44976 T^{3} + 924194 T^{4} - 44976 p^{2} T^{5} + 1152 p^{4} T^{6} - 48 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 96 T + 4608 T^{2} - 281184 T^{3} + 16639682 T^{4} - 281184 p^{2} T^{5} + 4608 p^{4} T^{6} - 96 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 100 T + 5000 T^{2} + 17100 T^{3} - 6900562 T^{4} + 17100 p^{2} T^{5} + 5000 p^{4} T^{6} + 100 p^{6} T^{7} + p^{8} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 6562 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 48 T + 6482 T^{2} + 48 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 10128 T^{3} - 14067358 T^{4} + 10128 p^{2} T^{5} + 128 p^{4} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 32 T + 10242 T^{2} + 32 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 188 T + 17672 T^{2} - 1796340 T^{3} + 164736974 T^{4} - 1796340 p^{2} T^{5} + 17672 p^{4} T^{6} - 188 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 7772 T^{2} + 83656134 T^{4} + 7772 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 192 T + 18432 T^{2} - 1755840 T^{3} + 162172514 T^{4} - 1755840 p^{2} T^{5} + 18432 p^{4} T^{6} - 192 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 14468 T^{2} + 151051974 T^{4} - 14468 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 132 T + 8712 T^{2} + 895884 T^{3} + 85251854 T^{4} + 895884 p^{2} T^{5} + 8712 p^{4} T^{6} + 132 p^{6} T^{7} + p^{8} T^{8} \) |
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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
−6.47104725924836479554086992557, −6.45707182855921860463375549416, −6.40081315732320499601694244560, −6.38888273755189145671684134026, −6.17739106006608488847600468770, −5.69348605993407217067187716764, −5.47558928234450662581885923122, −5.29655567961762311232700818379, −4.83916789148822260937104412237, −4.49908606880532223348663869202, −4.38508936698634136366505712237, −4.32346366783776879177632365508, −4.04643032586110634485668800596, −3.72406963077434138017780801432, −3.68444878202062920746964069214, −2.88199581086059563100278810869, −2.86748045631290263245407436353, −2.81245487252358025399883340755, −2.53642602730959678550735837048, −2.06678857675431633446425301179, −1.96272098731485425765894182332, −1.25490119068667392464924193983, −0.916491925422925666755536836958, −0.63664434828872850653594233294, −0.18174165377111704725271937790,
0.18174165377111704725271937790, 0.63664434828872850653594233294, 0.916491925422925666755536836958, 1.25490119068667392464924193983, 1.96272098731485425765894182332, 2.06678857675431633446425301179, 2.53642602730959678550735837048, 2.81245487252358025399883340755, 2.86748045631290263245407436353, 2.88199581086059563100278810869, 3.68444878202062920746964069214, 3.72406963077434138017780801432, 4.04643032586110634485668800596, 4.32346366783776879177632365508, 4.38508936698634136366505712237, 4.49908606880532223348663869202, 4.83916789148822260937104412237, 5.29655567961762311232700818379, 5.47558928234450662581885923122, 5.69348605993407217067187716764, 6.17739106006608488847600468770, 6.38888273755189145671684134026, 6.40081315732320499601694244560, 6.45707182855921860463375549416, 6.47104725924836479554086992557
