| L(s) = 1 | + 2·2-s + 4-s − 2·5-s − 2·8-s − 4·10-s + 2·11-s − 8·13-s − 4·16-s + 2·17-s − 4·19-s − 2·20-s + 4·22-s + 8·23-s + 4·25-s − 16·26-s − 20·29-s + 4·31-s − 2·32-s + 4·34-s + 8·37-s − 8·38-s + 4·40-s − 12·41-s + 20·43-s + 2·44-s + 16·46-s − 6·47-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s − 0.894·5-s − 0.707·8-s − 1.26·10-s + 0.603·11-s − 2.21·13-s − 16-s + 0.485·17-s − 0.917·19-s − 0.447·20-s + 0.852·22-s + 1.66·23-s + 4/5·25-s − 3.13·26-s − 3.71·29-s + 0.718·31-s − 0.353·32-s + 0.685·34-s + 1.31·37-s − 1.29·38-s + 0.632·40-s − 1.87·41-s + 3.04·43-s + 0.301·44-s + 2.35·46-s − 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.828175104\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.828175104\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 2 T - 12 T^{3} - 29 T^{4} - 12 p T^{5} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 2 T - 12 T^{2} + 12 T^{3} + 91 T^{4} + 12 p T^{5} - 12 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 4 T + 23 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 2 T - 24 T^{2} + 12 T^{3} + 427 T^{4} + 12 p T^{5} - 24 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 8 T + 30 T^{2} + 96 T^{3} - 845 T^{4} + 96 p T^{5} + 30 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 10 T + 76 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 4 T - 43 T^{2} + 12 T^{3} + 2024 T^{4} + 12 p T^{5} - 43 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \) |
| 41 | $D_{4}$ | \( ( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{4} \) |
| 47 | $D_4\times C_2$ | \( 1 + 6 T - 4 T^{2} - 324 T^{3} - 2301 T^{4} - 324 p T^{5} - 4 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 22 T + 252 T^{2} - 2508 T^{3} + 21787 T^{4} - 2508 p T^{5} + 252 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 20 T + 185 T^{2} + 1860 T^{3} + 18104 T^{4} + 1860 p T^{5} + 185 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 6 T - 79 T^{2} - 114 T^{3} + 6324 T^{4} - 114 p T^{5} - 79 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 26 T + 304 T^{2} + 26 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^3$ | \( 1 - 34 T^{2} - 4173 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 4 T + 29 T^{2} + 684 T^{3} - 7336 T^{4} + 684 p T^{5} + 29 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 16 T + 202 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 6 T - 88 T^{2} - 324 T^{3} + 4251 T^{4} - 324 p T^{5} - 88 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 6 T + 91 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−8.296726630189243408480826634837, −7.62745298964422116029932758250, −7.54679335060942791678207626479, −7.48508814832582750000513865064, −7.38541746279037188231615354084, −6.95644672595677303560922918027, −6.93515949454679271379175924539, −6.29555167016919467141217585247, −6.01484335420599004594830298439, −5.89228674916507380439529391223, −5.73625978484083783714330917461, −5.15945834796982949152461143145, −5.08921653723047640987059495213, −4.88897227986472045382410632931, −4.30397081887324486776967804390, −4.26986174262211638135468398266, −4.23653870630519491705674900881, −3.60218702640919767909995378601, −3.43248451364571585966067193708, −3.11318357093167785124822727914, −2.64666046806064661209124295917, −2.35212087482632789877335464207, −2.05215780553305001286438904907, −1.24359319753871805406851490265, −0.44371697773959580542864949800,
0.44371697773959580542864949800, 1.24359319753871805406851490265, 2.05215780553305001286438904907, 2.35212087482632789877335464207, 2.64666046806064661209124295917, 3.11318357093167785124822727914, 3.43248451364571585966067193708, 3.60218702640919767909995378601, 4.23653870630519491705674900881, 4.26986174262211638135468398266, 4.30397081887324486776967804390, 4.88897227986472045382410632931, 5.08921653723047640987059495213, 5.15945834796982949152461143145, 5.73625978484083783714330917461, 5.89228674916507380439529391223, 6.01484335420599004594830298439, 6.29555167016919467141217585247, 6.93515949454679271379175924539, 6.95644672595677303560922918027, 7.38541746279037188231615354084, 7.48508814832582750000513865064, 7.54679335060942791678207626479, 7.62745298964422116029932758250, 8.296726630189243408480826634837
