| L(s) = 1 | + 4·2-s + 10·4-s + 3·5-s + 20·8-s + 12·10-s + 3·11-s − 4·13-s + 35·16-s − 3·17-s − 10·19-s + 30·20-s + 12·22-s − 9·23-s + 4·25-s − 16·26-s − 6·29-s + 8·31-s + 56·32-s − 12·34-s − 4·37-s − 40·38-s + 60·40-s + 15·41-s − 43-s + 30·44-s − 36·46-s + 16·50-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 5·4-s + 1.34·5-s + 7.07·8-s + 3.79·10-s + 0.904·11-s − 1.10·13-s + 35/4·16-s − 0.727·17-s − 2.29·19-s + 6.70·20-s + 2.55·22-s − 1.87·23-s + 4/5·25-s − 3.13·26-s − 1.11·29-s + 1.43·31-s + 9.89·32-s − 2.05·34-s − 0.657·37-s − 6.48·38-s + 9.48·40-s + 2.34·41-s − 0.152·43-s + 4.52·44-s − 5.30·46-s + 2.26·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(42.98275594\) |
| \(L(\frac12)\) |
\(\approx\) |
\(42.98275594\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$$\times$$C_2^2$ | \( ( 1 - 3 T + p T^{2} )^{2}( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} ) \) |
| 11 | $D_4\times C_2$ | \( 1 - 3 T - 7 T^{2} + 18 T^{3} + 36 T^{4} + 18 p T^{5} - 7 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 3 T - 19 T^{2} - 18 T^{3} + 342 T^{4} - 18 p T^{5} - 19 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 + 9 T + p T^{2} )^{2}( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} ) \) |
| 29 | $D_4\times C_2$ | \( 1 + 6 T + 2 T^{2} - 144 T^{3} - 729 T^{4} - 144 p T^{5} + 2 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 15 T + 95 T^{2} - 720 T^{3} + 5994 T^{4} - 720 p T^{5} + 95 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + T - 11 T^{2} - 74 T^{3} - 1748 T^{4} - 74 p T^{5} - 11 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $D_4\times C_2$ | \( 1 + 6 T - 46 T^{2} - 144 T^{3} + 2007 T^{4} - 144 p T^{5} - 46 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 3 T + 46 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 11 T + 78 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 3 T + 70 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 7 T - 35 T^{2} - 434 T^{3} - 1850 T^{4} - 434 p T^{5} - 35 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 7 T + 96 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 12 T + 74 T^{2} + 1152 T^{3} - 13941 T^{4} + 1152 p T^{5} + 74 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 18 T + 98 T^{2} - 864 T^{3} + 14319 T^{4} - 864 p T^{5} + 98 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + T - 119 T^{2} - 74 T^{3} + 4894 T^{4} - 74 p T^{5} - 119 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.16349442474585514889500394803, −6.14557249483597741275419242083, −5.97692388736682339388444940426, −5.49382047581327535156271645130, −5.39012109277976157242743930542, −5.38169671979976514759539887121, −4.99048397294320159524463810007, −4.88374947389448649423613455752, −4.52129708674388724569594234961, −4.34519895255514956191437524721, −4.20502679160136556720991990647, −4.03871349875158789621991271476, −4.03228505403508083866094026680, −3.52789383091931251870990051728, −3.42234928239996472665550013074, −2.99689629495809793551049072367, −2.79342185460692474843589073734, −2.56567233978170825614315212305, −2.40207584929851166415748235320, −1.93677687423672463363214389870, −1.91947390343428096462302538737, −1.83582728903719678773161315886, −1.56139984715247776059631948601, −0.72715680072186635043737100441, −0.52656568193688303610039582560,
0.52656568193688303610039582560, 0.72715680072186635043737100441, 1.56139984715247776059631948601, 1.83582728903719678773161315886, 1.91947390343428096462302538737, 1.93677687423672463363214389870, 2.40207584929851166415748235320, 2.56567233978170825614315212305, 2.79342185460692474843589073734, 2.99689629495809793551049072367, 3.42234928239996472665550013074, 3.52789383091931251870990051728, 4.03228505403508083866094026680, 4.03871349875158789621991271476, 4.20502679160136556720991990647, 4.34519895255514956191437524721, 4.52129708674388724569594234961, 4.88374947389448649423613455752, 4.99048397294320159524463810007, 5.38169671979976514759539887121, 5.39012109277976157242743930542, 5.49382047581327535156271645130, 5.97692388736682339388444940426, 6.14557249483597741275419242083, 6.16349442474585514889500394803
