| L(s) = 1 | − 4·2-s + 4·4-s − 12·5-s − 16·7-s + 16·8-s + 48·10-s − 36·11-s − 10·13-s + 64·14-s − 64·16-s + 240·17-s − 16·19-s − 48·20-s + 144·22-s − 180·23-s + 259·25-s + 40·26-s − 64·28-s − 324·29-s + 248·31-s + 64·32-s − 960·34-s + 192·35-s − 868·37-s + 64·38-s − 192·40-s − 192·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s − 1.07·5-s − 0.863·7-s + 0.707·8-s + 1.51·10-s − 0.986·11-s − 0.213·13-s + 1.22·14-s − 16-s + 3.42·17-s − 0.193·19-s − 0.536·20-s + 1.39·22-s − 1.63·23-s + 2.07·25-s + 0.301·26-s − 0.431·28-s − 2.07·29-s + 1.43·31-s + 0.353·32-s − 4.84·34-s + 0.927·35-s − 3.85·37-s + 0.273·38-s − 0.758·40-s − 0.731·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.04176682411\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04176682411\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 12 T - 23 p T^{2} + 108 T^{3} + 33456 T^{4} + 108 p^{3} T^{5} - 23 p^{7} T^{6} + 12 p^{9} T^{7} + p^{12} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 16 T - 386 T^{2} - 704 T^{3} + 206707 T^{4} - 704 p^{3} T^{5} - 386 p^{6} T^{6} + 16 p^{9} T^{7} + p^{12} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 36 T + 38 T^{2} - 50544 T^{3} - 1913973 T^{4} - 50544 p^{3} T^{5} + 38 p^{6} T^{6} + 36 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 10 T + 973 T^{2} - 52670 T^{3} - 4284380 T^{4} - 52670 p^{3} T^{5} + 973 p^{6} T^{6} + 10 p^{9} T^{7} + p^{12} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 120 T + 10159 T^{2} - 120 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 8 T + 666 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 180 T + 398 T^{2} + 1380240 T^{3} + 481881315 T^{4} + 1380240 p^{3} T^{5} + 398 p^{6} T^{6} + 180 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 324 T + 31277 T^{2} + 8074404 T^{3} + 2276459616 T^{4} + 8074404 p^{3} T^{5} + 31277 p^{6} T^{6} + 324 p^{9} T^{7} + p^{12} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 8 p T + 21538 T^{2} + 156928 p T^{3} - 1122488189 T^{4} + 156928 p^{4} T^{5} + 21538 p^{6} T^{6} - 8 p^{10} T^{7} + p^{12} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 434 T + 148287 T^{2} + 434 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 192 T - 57922 T^{2} - 8266752 T^{3} + 1693577811 T^{4} - 8266752 p^{3} T^{5} - 57922 p^{6} T^{6} + 192 p^{9} T^{7} + p^{12} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 608 T + 119206 T^{2} - 55597952 T^{3} + 27016971067 T^{4} - 55597952 p^{3} T^{5} + 119206 p^{6} T^{6} - 608 p^{9} T^{7} + p^{12} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 384 T - 62062 T^{2} - 718848 T^{3} + 17809252707 T^{4} - 718848 p^{3} T^{5} - 62062 p^{6} T^{6} - 384 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 408 T + 183418 T^{2} + 408 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 1008 T + 378938 T^{2} - 228178944 T^{3} + 155070797979 T^{4} - 228178944 p^{3} T^{5} + 378938 p^{6} T^{6} - 1008 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 742 T - 32291 T^{2} + 1567846 p T^{3} + 43867780 p^{2} T^{4} + 1567846 p^{4} T^{5} - 32291 p^{6} T^{6} + 742 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 104 T - 79226 T^{2} + 53194336 T^{3} - 85603873013 T^{4} + 53194336 p^{3} T^{5} - 79226 p^{6} T^{6} - 104 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 1140 T + 1038994 T^{2} + 1140 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 850 T + 709827 T^{2} - 850 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 440 T - 773378 T^{2} + 8404000 T^{3} + 596506646563 T^{4} + 8404000 p^{3} T^{5} - 773378 p^{6} T^{6} - 440 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 264 T - 994102 T^{2} - 21060864 T^{3} + 764874544683 T^{4} - 21060864 p^{3} T^{5} - 994102 p^{6} T^{6} + 264 p^{9} T^{7} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 768 T + 1343527 T^{2} - 768 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 764 T - 1302902 T^{2} - 46796528 T^{3} + 2193556429267 T^{4} - 46796528 p^{3} T^{5} - 1302902 p^{6} T^{6} - 764 p^{9} T^{7} + p^{12} T^{8} \) |
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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
−8.779804123016023083445843169348, −8.774865336778654050781247808542, −8.700796253027268940212411594364, −7.83866381974852619143053490592, −7.80033146567465671010915497835, −7.78986703347105247686208000061, −7.39312989547036953585673406555, −7.35621031450140624593156658227, −6.92455721112935796467988800521, −6.42542566628063515798762971333, −6.08793980701457453231757441152, −5.80800665698393674103877175714, −5.49304014983334770410458792059, −5.08416857611124138766289464974, −4.93957827927447405269927582112, −4.45319584910810681867440777759, −3.87053034668865712966297529354, −3.51343631637813095849762547799, −3.47134011785564849430654757658, −3.08378712254879754004793438600, −2.34555889312077755416378264004, −1.99866107777991359429592635759, −1.13337653420785352725554600093, −0.883878082024610506740370568450, −0.080831565512253468667611489821,
0.080831565512253468667611489821, 0.883878082024610506740370568450, 1.13337653420785352725554600093, 1.99866107777991359429592635759, 2.34555889312077755416378264004, 3.08378712254879754004793438600, 3.47134011785564849430654757658, 3.51343631637813095849762547799, 3.87053034668865712966297529354, 4.45319584910810681867440777759, 4.93957827927447405269927582112, 5.08416857611124138766289464974, 5.49304014983334770410458792059, 5.80800665698393674103877175714, 6.08793980701457453231757441152, 6.42542566628063515798762971333, 6.92455721112935796467988800521, 7.35621031450140624593156658227, 7.39312989547036953585673406555, 7.78986703347105247686208000061, 7.80033146567465671010915497835, 7.83866381974852619143053490592, 8.700796253027268940212411594364, 8.774865336778654050781247808542, 8.779804123016023083445843169348
