| L(s) = 1 | − 8·5-s + 30·7-s + 46·11-s − 92·13-s + 44·17-s − 172·19-s + 58·23-s + 65·25-s − 108·29-s − 136·31-s − 240·35-s − 360·37-s − 672·41-s + 70·43-s − 852·47-s + 710·49-s + 1.33e3·53-s − 368·55-s − 548·59-s − 284·61-s + 736·65-s − 1.16e3·67-s − 1.61e3·71-s − 3.02e3·73-s + 1.38e3·77-s + 22·79-s − 1.54e3·83-s + ⋯ |
| L(s) = 1 | − 0.715·5-s + 1.61·7-s + 1.26·11-s − 1.96·13-s + 0.627·17-s − 2.07·19-s + 0.525·23-s + 0.519·25-s − 0.691·29-s − 0.787·31-s − 1.15·35-s − 1.59·37-s − 2.55·41-s + 0.248·43-s − 2.64·47-s + 2.06·49-s + 3.46·53-s − 0.902·55-s − 1.20·59-s − 0.596·61-s + 1.40·65-s − 2.11·67-s − 2.69·71-s − 4.84·73-s + 2.04·77-s + 0.0313·79-s − 2.03·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4315000357\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4315000357\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 8 T - T^{2} - 296 p T^{3} - 776 p^{2} T^{4} - 296 p^{4} T^{5} - p^{6} T^{6} + 8 p^{9} T^{7} + p^{12} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 30 T + 190 T^{2} - 720 T^{3} + 77751 T^{4} - 720 p^{3} T^{5} + 190 p^{6} T^{6} - 30 p^{9} T^{7} + p^{12} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 46 T + 734 T^{2} + 58880 T^{3} - 2678033 T^{4} + 58880 p^{3} T^{5} + 734 p^{6} T^{6} - 46 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 92 T + 2155 T^{2} + 176180 T^{3} + 16381264 T^{4} + 176180 p^{3} T^{5} + 2155 p^{6} T^{6} + 92 p^{9} T^{7} + p^{12} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 22 T - 2917 T^{2} - 22 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 86 T + 10542 T^{2} + 86 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 58 T - 21610 T^{2} - 37120 T^{3} + 434734999 T^{4} - 37120 p^{3} T^{5} - 21610 p^{6} T^{6} - 58 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 108 T - 39829 T^{2} + 293220 T^{3} + 1772232432 T^{4} + 293220 p^{3} T^{5} - 39829 p^{6} T^{6} + 108 p^{9} T^{7} + p^{12} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 136 T + 5746 T^{2} - 6369152 T^{3} - 1275955517 T^{4} - 6369152 p^{3} T^{5} + 5746 p^{6} T^{6} + 136 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 180 T + 109205 T^{2} + 180 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 672 T + 201650 T^{2} + 75325824 T^{3} + 26593279251 T^{4} + 75325824 p^{3} T^{5} + 201650 p^{6} T^{6} + 672 p^{9} T^{7} + p^{12} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 70 T - 49010 T^{2} + 7357280 T^{3} - 3804659249 T^{4} + 7357280 p^{3} T^{5} - 49010 p^{6} T^{6} - 70 p^{9} T^{7} + p^{12} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 852 T + 337586 T^{2} + 153932544 T^{3} + 64646122803 T^{4} + 153932544 p^{3} T^{5} + 337586 p^{6} T^{6} + 852 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 668 T + 357854 T^{2} - 668 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 548 T - 178294 T^{2} + 37176320 T^{3} + 125199376747 T^{4} + 37176320 p^{3} T^{5} - 178294 p^{6} T^{6} + 548 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 284 T - 335381 T^{2} - 10770700 T^{3} + 103259031472 T^{4} - 10770700 p^{3} T^{5} - 335381 p^{6} T^{6} + 284 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 1162 T + 435478 T^{2} + 363984880 T^{3} + 340235016415 T^{4} + 363984880 p^{3} T^{5} + 435478 p^{6} T^{6} + 1162 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 806 T + 876422 T^{2} + 806 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 1510 T + 1282935 T^{2} + 1510 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 22 T - 647834 T^{2} + 7430720 T^{3} + 176990142727 T^{4} + 7430720 p^{3} T^{5} - 647834 p^{6} T^{6} - 22 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 1540 T + 925370 T^{2} + 466090240 T^{3} + 408590790331 T^{4} + 466090240 p^{3} T^{5} + 925370 p^{6} T^{6} + 1540 p^{9} T^{7} + p^{12} T^{8} \) |
| 89 | $C_2$ | \( ( 1 - 1323 T + p^{3} T^{2} )^{4} \) |
| 97 | $D_4\times C_2$ | \( 1 + 472 T - 1155758 T^{2} - 210891488 T^{3} + 864077099203 T^{4} - 210891488 p^{3} T^{5} - 1155758 p^{6} T^{6} + 472 p^{9} T^{7} + p^{12} T^{8} \) |
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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
−7.23770839781293046169612922084, −7.08870989300979748698456039599, −6.87493077085931540941451257785, −6.39610938623401255562133268234, −6.33031513981930267417000824037, −5.94171062991718319839567348816, −5.82883420692286476517917298729, −5.30925894305062834114222246171, −5.02489127424313060959225796348, −4.99380943344535363350738195824, −4.92506044821481245064456174046, −4.32919789987927628988843766250, −4.25463472175053198906267642229, −4.00865596334124051192012362529, −3.84905901404761364164469600272, −3.29852017618117473289401513329, −2.99979474802813970241092116733, −2.87889496293838905717336712905, −2.43486581361200335138747293271, −1.79522435663794948588792143268, −1.75840110417745940653581482331, −1.60962688880860810806467503402, −1.24831796389745582490219618672, −0.44765400869901656363301805326, −0.12109995951226200537184716676,
0.12109995951226200537184716676, 0.44765400869901656363301805326, 1.24831796389745582490219618672, 1.60962688880860810806467503402, 1.75840110417745940653581482331, 1.79522435663794948588792143268, 2.43486581361200335138747293271, 2.87889496293838905717336712905, 2.99979474802813970241092116733, 3.29852017618117473289401513329, 3.84905901404761364164469600272, 4.00865596334124051192012362529, 4.25463472175053198906267642229, 4.32919789987927628988843766250, 4.92506044821481245064456174046, 4.99380943344535363350738195824, 5.02489127424313060959225796348, 5.30925894305062834114222246171, 5.82883420692286476517917298729, 5.94171062991718319839567348816, 6.33031513981930267417000824037, 6.39610938623401255562133268234, 6.87493077085931540941451257785, 7.08870989300979748698456039599, 7.23770839781293046169612922084
