| L(s) = 1 | + 18·5-s − 2·7-s + 6·9-s + 18·11-s + 10·13-s − 40·19-s − 18·23-s + 139·25-s − 18·29-s − 38·31-s − 36·35-s − 128·37-s − 126·41-s − 46·43-s + 108·45-s − 54·47-s + 45·49-s + 324·55-s + 126·59-s − 62·61-s − 12·63-s + 180·65-s − 106·67-s − 208·73-s − 36·77-s − 14·79-s − 45·81-s + ⋯ |
| L(s) = 1 | + 18/5·5-s − 2/7·7-s + 2/3·9-s + 1.63·11-s + 0.769·13-s − 2.10·19-s − 0.782·23-s + 5.55·25-s − 0.620·29-s − 1.22·31-s − 1.02·35-s − 3.45·37-s − 3.07·41-s − 1.06·43-s + 12/5·45-s − 1.14·47-s + 0.918·49-s + 5.89·55-s + 2.13·59-s − 1.01·61-s − 0.190·63-s + 2.76·65-s − 1.58·67-s − 2.84·73-s − 0.467·77-s − 0.177·79-s − 5/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.07120322315\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07120322315\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 - 2 p T^{2} + p^{4} T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 - 9 T + 52 T^{2} - 9 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 + 2 T - 41 T^{2} - 106 T^{3} - 572 T^{4} - 106 p^{2} T^{5} - 41 p^{4} T^{6} + 2 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 18 T + 359 T^{2} - 4518 T^{3} + 61428 T^{4} - 4518 p^{2} T^{5} + 359 p^{4} T^{6} - 18 p^{6} T^{7} + p^{8} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 10 T - 47 T^{2} + 1910 T^{3} - 23852 T^{4} + 1910 p^{2} T^{5} - 47 p^{4} T^{6} - 10 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 796 T^{2} + 294342 T^{4} - 796 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 20 T + 606 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 18 T + 1175 T^{2} + 19206 T^{3} + 915780 T^{4} + 19206 p^{2} T^{5} + 1175 p^{4} T^{6} + 18 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 18 T + 1745 T^{2} + 29466 T^{3} + 2063316 T^{4} + 29466 p^{2} T^{5} + 1745 p^{4} T^{6} + 18 p^{6} T^{7} + p^{8} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 38 T - 353 T^{2} - 4750 T^{3} + 918004 T^{4} - 4750 p^{2} T^{5} - 353 p^{4} T^{6} + 38 p^{6} T^{7} + p^{8} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 64 T + 3546 T^{2} + 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 126 T + 9329 T^{2} + 508662 T^{3} + 22367460 T^{4} + 508662 p^{2} T^{5} + 9329 p^{4} T^{6} + 126 p^{6} T^{7} + p^{8} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 46 T - 1625 T^{2} + 46 p T^{3} + 3604 p^{2} T^{4} + 46 p^{3} T^{5} - 1625 p^{4} T^{6} + 46 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 54 T + 4751 T^{2} + 204066 T^{3} + 11548308 T^{4} + 204066 p^{2} T^{5} + 4751 p^{4} T^{6} + 54 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 2236 T^{2} - 2409114 T^{4} - 2236 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 126 T + 10535 T^{2} - 660618 T^{3} + 33793140 T^{4} - 660618 p^{2} T^{5} + 10535 p^{4} T^{6} - 126 p^{6} T^{7} + p^{8} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 62 T - 2615 T^{2} - 60946 T^{3} + 13569316 T^{4} - 60946 p^{2} T^{5} - 2615 p^{4} T^{6} + 62 p^{6} T^{7} + p^{8} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 106 T - 65 T^{2} + 246238 T^{3} + 57123076 T^{4} + 246238 p^{2} T^{5} - 65 p^{4} T^{6} + 106 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 12460 T^{2} + 77194662 T^{4} - 12460 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 104 T + 11418 T^{2} + 104 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 14 T - 10985 T^{2} - 18214 T^{3} + 84841444 T^{4} - 18214 p^{2} T^{5} - 10985 p^{4} T^{6} + 14 p^{6} T^{7} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 378 T + 72863 T^{2} + 9538830 T^{3} + 917456196 T^{4} + 9538830 p^{2} T^{5} + 72863 p^{4} T^{6} + 378 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 8860 T^{2} + 51019782 T^{4} - 8860 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 14 T - 8087 T^{2} + 147490 T^{3} - 21765356 T^{4} + 147490 p^{2} T^{5} - 8087 p^{4} T^{6} - 14 p^{6} T^{7} + p^{8} T^{8} \) |
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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
−7.12655971716861011197954243094, −7.11198830769524451160745770808, −6.94023902578078002041199752490, −6.85176838554732960505153757443, −6.55067386363148248894577142827, −6.18174945966648314597604852453, −5.98653561837124977582850622330, −5.90584200079107919716061371666, −5.62391994042219235850222016039, −5.55500507835713216341604628577, −5.23662732531748738294458163012, −4.74302383927982544278648517969, −4.54303079371355204684569781581, −4.38182885062435956841225093432, −3.74601650948228487231464284935, −3.66324046590795350632344913595, −3.54244097487525556179003260428, −2.83094626467658825976903843923, −2.79502802025449987847132248846, −1.95189839422647724100599070588, −1.83076158395789606916358366616, −1.71357653967474329733869377495, −1.69087430835441442655155763358, −1.29900846337564612819518336076, −0.03397242950084387879984588801,
0.03397242950084387879984588801, 1.29900846337564612819518336076, 1.69087430835441442655155763358, 1.71357653967474329733869377495, 1.83076158395789606916358366616, 1.95189839422647724100599070588, 2.79502802025449987847132248846, 2.83094626467658825976903843923, 3.54244097487525556179003260428, 3.66324046590795350632344913595, 3.74601650948228487231464284935, 4.38182885062435956841225093432, 4.54303079371355204684569781581, 4.74302383927982544278648517969, 5.23662732531748738294458163012, 5.55500507835713216341604628577, 5.62391994042219235850222016039, 5.90584200079107919716061371666, 5.98653561837124977582850622330, 6.18174945966648314597604852453, 6.55067386363148248894577142827, 6.85176838554732960505153757443, 6.94023902578078002041199752490, 7.11198830769524451160745770808, 7.12655971716861011197954243094
