| L(s) = 1 | + 2-s − 12·3-s − 8·4-s − 9·5-s − 12·6-s − 35·7-s − 24·8-s + 90·9-s − 9·10-s + 47·11-s + 96·12-s + 85·13-s − 35·14-s + 108·15-s − 3·16-s + 90·18-s − 29·19-s + 72·20-s + 420·21-s + 47·22-s − 60·23-s + 288·24-s − 173·25-s + 85·26-s − 540·27-s + 280·28-s + 39·29-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 2.30·3-s − 4-s − 0.804·5-s − 0.816·6-s − 1.88·7-s − 1.06·8-s + 10/3·9-s − 0.284·10-s + 1.28·11-s + 2.30·12-s + 1.81·13-s − 0.668·14-s + 1.85·15-s − 0.0468·16-s + 1.17·18-s − 0.350·19-s + 0.804·20-s + 4.36·21-s + 0.455·22-s − 0.543·23-s + 2.44·24-s − 1.38·25-s + 0.641·26-s − 3.84·27-s + 1.88·28-s + 0.249·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.07153034090\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07153034090\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + p T )^{4} \) |
| 17 | | \( 1 \) |
| good | 2 | $C_2 \wr S_4$ | \( 1 - T + 9 T^{2} + 7 T^{3} + 11 p^{2} T^{4} + 7 p^{3} T^{5} + 9 p^{6} T^{6} - p^{9} T^{7} + p^{12} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 + 9 T + 254 T^{2} + 423 T^{3} + 25994 T^{4} + 423 p^{3} T^{5} + 254 p^{6} T^{6} + 9 p^{9} T^{7} + p^{12} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 + 5 p T + 1331 T^{2} + 29208 T^{3} + 639932 T^{4} + 29208 p^{3} T^{5} + 1331 p^{6} T^{6} + 5 p^{10} T^{7} + p^{12} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 47 T + 2608 T^{2} - 120987 T^{3} + 5766302 T^{4} - 120987 p^{3} T^{5} + 2608 p^{6} T^{6} - 47 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 - 85 T + 5513 T^{2} - 132234 T^{3} + 5981062 T^{4} - 132234 p^{3} T^{5} + 5513 p^{6} T^{6} - 85 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 29 T + 13523 T^{2} - 20244 p T^{3} + 75083300 T^{4} - 20244 p^{4} T^{5} + 13523 p^{6} T^{6} + 29 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 60 T + 17720 T^{2} + 1275660 T^{3} + 382421006 T^{4} + 1275660 p^{3} T^{5} + 17720 p^{6} T^{6} + 60 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 39 T + 22284 T^{2} - 174621 T^{3} + 697436422 T^{4} - 174621 p^{3} T^{5} + 22284 p^{6} T^{6} - 39 p^{9} T^{7} + p^{12} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 342 T + 76620 T^{2} + 11555576 T^{3} + 2166641157 T^{4} + 11555576 p^{3} T^{5} + 76620 p^{6} T^{6} + 342 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 549 T + 292215 T^{2} + 88242794 T^{3} + 24527184420 T^{4} + 88242794 p^{3} T^{5} + 292215 p^{6} T^{6} + 549 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 102 T + 233424 T^{2} - 16736730 T^{3} + 22580260702 T^{4} - 16736730 p^{3} T^{5} + 233424 p^{6} T^{6} - 102 p^{9} T^{7} + p^{12} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 797 T + 411947 T^{2} - 149795340 T^{3} + 47337992636 T^{4} - 149795340 p^{3} T^{5} + 411947 p^{6} T^{6} - 797 p^{9} T^{7} + p^{12} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 130 T + 110812 T^{2} - 56135910 T^{3} - 4457775610 T^{4} - 56135910 p^{3} T^{5} + 110812 p^{6} T^{6} + 130 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 47 T + 410284 T^{2} + 2537913 T^{3} + 81778738982 T^{4} + 2537913 p^{3} T^{5} + 410284 p^{6} T^{6} + 47 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 549 T + 876984 T^{2} - 335147577 T^{3} + 275814575086 T^{4} - 335147577 p^{3} T^{5} + 876984 p^{6} T^{6} - 549 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 1175 T + 1300133 T^{2} + 847667550 T^{3} + 489079033138 T^{4} + 847667550 p^{3} T^{5} + 1300133 p^{6} T^{6} + 1175 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 1465 T + 1709677 T^{2} - 1350002212 T^{3} + 841967699422 T^{4} - 1350002212 p^{3} T^{5} + 1709677 p^{6} T^{6} - 1465 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 830 T + 789084 T^{2} + 622153702 T^{3} + 335665885094 T^{4} + 622153702 p^{3} T^{5} + 789084 p^{6} T^{6} + 830 p^{9} T^{7} + p^{12} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 1265 T + 1890366 T^{2} - 1457351847 T^{3} + 1160366421538 T^{4} - 1457351847 p^{3} T^{5} + 1890366 p^{6} T^{6} - 1265 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 165 T + 1327036 T^{2} - 319061169 T^{3} + 859091519814 T^{4} - 319061169 p^{3} T^{5} + 1327036 p^{6} T^{6} - 165 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 1814 T + 1851660 T^{2} + 949344070 T^{3} + 548147242838 T^{4} + 949344070 p^{3} T^{5} + 1851660 p^{6} T^{6} + 1814 p^{9} T^{7} + p^{12} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 1938 T + 3434456 T^{2} - 3565166622 T^{3} + 3612751901102 T^{4} - 3565166622 p^{3} T^{5} + 3434456 p^{6} T^{6} - 1938 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 2954 T + 5863700 T^{2} + 7891887300 T^{3} + 8666715233413 T^{4} + 7891887300 p^{3} T^{5} + 5863700 p^{6} T^{6} + 2954 p^{9} T^{7} + p^{12} T^{8} \) |
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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
−6.80057692709422321435884398261, −6.57129214667124722007299820332, −6.17601899321999053969563088845, −6.16839533160315592464178820179, −6.00491246745158157862317879464, −5.90827047993528931097139748004, −5.44857387613672643547368280549, −5.31043835716853771963338318773, −5.27723636043273141489640138973, −4.49839658129316028247490392548, −4.46428128400703077373382179826, −4.44906657350861140522410876751, −4.01509327835506274070020722466, −3.68533928771034743473322061576, −3.52832361218103052151873446786, −3.50962412419058390715721608632, −3.31881446715921978574992632373, −2.72965653374639346986331205903, −2.01719938353204761671934518986, −2.00452814729683018286633923154, −1.50462515067122332730292076707, −1.05932609296340146512377304434, −0.78055624942054051629420021079, −0.34180945324770858050881548313, −0.10099313425919209192254729945,
0.10099313425919209192254729945, 0.34180945324770858050881548313, 0.78055624942054051629420021079, 1.05932609296340146512377304434, 1.50462515067122332730292076707, 2.00452814729683018286633923154, 2.01719938353204761671934518986, 2.72965653374639346986331205903, 3.31881446715921978574992632373, 3.50962412419058390715721608632, 3.52832361218103052151873446786, 3.68533928771034743473322061576, 4.01509327835506274070020722466, 4.44906657350861140522410876751, 4.46428128400703077373382179826, 4.49839658129316028247490392548, 5.27723636043273141489640138973, 5.31043835716853771963338318773, 5.44857387613672643547368280549, 5.90827047993528931097139748004, 6.00491246745158157862317879464, 6.16839533160315592464178820179, 6.17601899321999053969563088845, 6.57129214667124722007299820332, 6.80057692709422321435884398261
