| L(s) = 1 | − 2·2-s − 6·3-s + 15·4-s + 20·5-s + 12·6-s − 50·8-s + 9·9-s − 40·10-s + 20·11-s − 90·12-s − 208·13-s − 120·15-s + 132·16-s + 116·17-s − 18·18-s + 192·19-s + 300·20-s − 40·22-s − 28·23-s + 300·24-s + 252·25-s + 416·26-s + 54·27-s + 592·29-s + 240·30-s − 104·31-s − 510·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 15/8·4-s + 1.78·5-s + 0.816·6-s − 2.20·8-s + 1/3·9-s − 1.26·10-s + 0.548·11-s − 2.16·12-s − 4.43·13-s − 2.06·15-s + 2.06·16-s + 1.65·17-s − 0.235·18-s + 2.31·19-s + 3.35·20-s − 0.387·22-s − 0.253·23-s + 2.55·24-s + 2.01·25-s + 3.13·26-s + 0.384·27-s + 3.79·29-s + 1.46·30-s − 0.602·31-s − 2.81·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.238484536\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.238484536\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 + p T - 11 T^{2} - p T^{3} + 129 T^{4} - p^{4} T^{5} - 11 p^{6} T^{6} + p^{10} T^{7} + p^{12} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 - 4 p T + 148 T^{2} - 8 p T^{3} - 2121 T^{4} - 8 p^{4} T^{5} + 148 p^{6} T^{6} - 4 p^{10} T^{7} + p^{12} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 20 T - 10 p^{2} T^{2} + 21040 T^{3} + 288139 T^{4} + 21040 p^{3} T^{5} - 10 p^{8} T^{6} - 20 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 8 p T + 5848 T^{2} + 8 p^{4} T^{3} + p^{6} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 116 T + 4316 T^{2} + 79576 T^{3} - 6707297 T^{4} + 79576 p^{3} T^{5} + 4316 p^{6} T^{6} - 116 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 192 T + 14898 T^{2} - 1583616 T^{3} + 182609099 T^{4} - 1583616 p^{3} T^{5} + 14898 p^{6} T^{6} - 192 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 28 T - 22178 T^{2} - 38416 T^{3} + 369678627 T^{4} - 38416 p^{3} T^{5} - 22178 p^{6} T^{6} + 28 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 296 T + 62994 T^{2} - 296 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 104 T - 46470 T^{2} - 238784 T^{3} + 2071962659 T^{4} - 238784 p^{3} T^{5} - 46470 p^{6} T^{6} + 104 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 248 T - 50570 T^{2} - 2670464 T^{3} + 6879492955 T^{4} - 2670464 p^{3} T^{5} - 50570 p^{6} T^{6} - 248 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 20 T + 42020 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 720 T + 268614 T^{2} + 720 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 96 T - 75734 T^{2} - 11778816 T^{3} - 4519545597 T^{4} - 11778816 p^{3} T^{5} - 75734 p^{6} T^{6} + 96 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 268 T + 15314 T^{2} - 64653392 T^{3} - 29663922677 T^{4} - 64653392 p^{3} T^{5} + 15314 p^{6} T^{6} + 268 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 616 T - 24014 T^{2} - 4489408 T^{3} + 42675213435 T^{4} - 4489408 p^{3} T^{5} - 24014 p^{6} T^{6} + 616 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 16 T - 453752 T^{2} - 736 T^{3} + 154544782567 T^{4} - 736 p^{3} T^{5} - 453752 p^{6} T^{6} - 16 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 144 T - 36822 T^{2} + 78331392 T^{3} - 93382080373 T^{4} + 78331392 p^{3} T^{5} - 36822 p^{6} T^{6} - 144 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 988 T + 852210 T^{2} - 988 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 104 T - 448320 T^{2} + 33165392 T^{3} + 55264032335 T^{4} + 33165392 p^{3} T^{5} - 448320 p^{6} T^{6} - 104 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 944 T - 206334 T^{2} - 105154048 T^{3} + 521988143075 T^{4} - 105154048 p^{3} T^{5} - 206334 p^{6} T^{6} - 944 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 1016 T + 1388838 T^{2} + 1016 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 388 T - 1067188 T^{2} - 74575928 T^{3} + 879761079727 T^{4} - 74575928 p^{3} T^{5} - 1067188 p^{6} T^{6} + 388 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 488 T + 1167280 T^{2} + 488 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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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
−9.429276254794841221700574429683, −9.017627183103384361066038825730, −8.316006429623685428968531499948, −8.293809095696283514516564338227, −8.083271500471040853015214583793, −7.38571021235192491930614930907, −7.30800577387066308504482766101, −7.05317882462747568893306197390, −6.83845940987919220056797863361, −6.47712107280020173955583384841, −6.35043217765508543467318021001, −5.84104207908973264954015298213, −5.77772507154076857588648576320, −5.19217255359235639264689126725, −4.91582320142464900671091672041, −4.82318156985108661696668126386, −4.81330454962934406631497936022, −3.27531479740885677910212422166, −3.08265946763753368443205584411, −3.06841816179699649541391407055, −2.41448032202452045622379673752, −2.09312749412864762960085492536, −1.68222898543569424510229853626, −0.943978668254005760938316291666, −0.45693484298887249970441401626,
0.45693484298887249970441401626, 0.943978668254005760938316291666, 1.68222898543569424510229853626, 2.09312749412864762960085492536, 2.41448032202452045622379673752, 3.06841816179699649541391407055, 3.08265946763753368443205584411, 3.27531479740885677910212422166, 4.81330454962934406631497936022, 4.82318156985108661696668126386, 4.91582320142464900671091672041, 5.19217255359235639264689126725, 5.77772507154076857588648576320, 5.84104207908973264954015298213, 6.35043217765508543467318021001, 6.47712107280020173955583384841, 6.83845940987919220056797863361, 7.05317882462747568893306197390, 7.30800577387066308504482766101, 7.38571021235192491930614930907, 8.083271500471040853015214583793, 8.293809095696283514516564338227, 8.316006429623685428968531499948, 9.017627183103384361066038825730, 9.429276254794841221700574429683
