| L(s) = 1 | − 2·4-s − 8·7-s + 8·11-s − 3·16-s − 8·17-s − 8·23-s + 16·28-s + 8·43-s − 16·44-s − 8·47-s + 28·49-s + 12·64-s + 16·68-s + 24·73-s − 64·77-s − 10·81-s − 24·83-s + 16·92-s − 32·101-s + 24·112-s + 64·119-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + ⋯ |
| L(s) = 1 | − 4-s − 3.02·7-s + 2.41·11-s − 3/4·16-s − 1.94·17-s − 1.66·23-s + 3.02·28-s + 1.21·43-s − 2.41·44-s − 1.16·47-s + 4·49-s + 3/2·64-s + 1.94·68-s + 2.80·73-s − 7.29·77-s − 1.11·81-s − 2.63·83-s + 1.66·92-s − 3.18·101-s + 2.26·112-s + 5.86·119-s − 0.363·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 19 | | \( 1 \) |
| good | 2 | $C_2^2 \wr C_2$ | \( 1 + p T^{2} + 7 T^{4} + p^{3} T^{6} + p^{4} T^{8} \) |
| 3 | $D_4\times C_2$ | \( 1 + 10 T^{4} + p^{4} T^{8} \) |
| 7 | $C_4$ | \( ( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 + 32 T^{2} + 522 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_4$ | \( ( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_4$ | \( ( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 + 36 T^{2} + 854 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 + 76 T^{2} + 3238 T^{4} + 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 + 96 T^{2} + 4394 T^{4} + 96 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $D_{4}$ | \( ( 1 - 4 T + 82 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 + 4 T + 26 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 + 128 T^{2} + 9322 T^{4} + 128 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 + 44 T^{2} + 5398 T^{4} + 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 114 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 + 192 T^{2} + 18122 T^{4} + 192 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 - 20 T^{2} + 9030 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2 \wr C_2$ | \( 1 + 124 T^{2} + 14278 T^{4} + 124 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 12 T + 194 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 + 20 T^{2} + 9670 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2 \wr C_2$ | \( 1 + 368 T^{2} + 52602 T^{4} + 368 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.02813772017780835294753965821, −5.52590116274528498389763865203, −5.52269211478100090097513552851, −5.18902973561342073603558013761, −5.12974184388643716235372962314, −4.73596958699551109845685360429, −4.62037667440260048112806831466, −4.40658647554188131274758573307, −4.20791704665193962222761352781, −4.08640538859869115364329445947, −3.98248720872125684332591481607, −3.79221192874926998740744211166, −3.61964320101115100932600576982, −3.43837781564133631463341469192, −3.33125013682997023943533880042, −2.84811175343825838842406199473, −2.83640241694790652602575109079, −2.51125621459489193678343230592, −2.38344046771761991318053005642, −2.17341168267226197715999493608, −1.90882176241246246318997668535, −1.59604930152559125842807188107, −1.23851865146180217784548947580, −1.06373579888924481848007889062, −0.848315027376495563502145530813, 0, 0, 0, 0,
0.848315027376495563502145530813, 1.06373579888924481848007889062, 1.23851865146180217784548947580, 1.59604930152559125842807188107, 1.90882176241246246318997668535, 2.17341168267226197715999493608, 2.38344046771761991318053005642, 2.51125621459489193678343230592, 2.83640241694790652602575109079, 2.84811175343825838842406199473, 3.33125013682997023943533880042, 3.43837781564133631463341469192, 3.61964320101115100932600576982, 3.79221192874926998740744211166, 3.98248720872125684332591481607, 4.08640538859869115364329445947, 4.20791704665193962222761352781, 4.40658647554188131274758573307, 4.62037667440260048112806831466, 4.73596958699551109845685360429, 5.12974184388643716235372962314, 5.18902973561342073603558013761, 5.52269211478100090097513552851, 5.52590116274528498389763865203, 6.02813772017780835294753965821
