| L(s) = 1 | + 2·2-s + 3-s + 2·4-s + 2·6-s + 2·7-s + 5·8-s + 3·9-s − 2·11-s + 2·12-s − 9·13-s + 4·14-s + 5·16-s − 8·17-s + 6·18-s − 5·19-s + 2·21-s − 4·22-s + 11·23-s + 5·24-s − 18·26-s + 4·28-s + 5·29-s + 3·31-s − 2·32-s − 2·33-s − 16·34-s + 6·36-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s + 0.816·6-s + 0.755·7-s + 1.76·8-s + 9-s − 0.603·11-s + 0.577·12-s − 2.49·13-s + 1.06·14-s + 5/4·16-s − 1.94·17-s + 1.41·18-s − 1.14·19-s + 0.436·21-s − 0.852·22-s + 2.29·23-s + 1.02·24-s − 3.53·26-s + 0.755·28-s + 0.928·29-s + 0.538·31-s − 0.353·32-s − 0.348·33-s − 2.74·34-s + 36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.992288518\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.992288518\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| good | 2 | $C_2^2:C_4$ | \( 1 - p T + p T^{2} - 5 T^{3} + 11 T^{4} - 5 p T^{5} + p^{3} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) |
| 3 | $C_4\times C_2$ | \( 1 - T - 2 T^{2} + 5 T^{3} + T^{4} + 5 p T^{5} - 2 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 - T + 13 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_4\times C_2$ | \( 1 + 2 T + 13 T^{2} + 34 T^{3} + 225 T^{4} + 34 p T^{5} + 13 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 + 9 T + 23 T^{2} + 15 T^{3} + 16 T^{4} + 15 p T^{5} + 23 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 + 8 T + 7 T^{2} - 110 T^{3} - 579 T^{4} - 110 p T^{5} + 7 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 + 5 T + 21 T^{2} + 145 T^{3} + 956 T^{4} + 145 p T^{5} + 21 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2:C_4$ | \( 1 - 11 T + 28 T^{2} + 245 T^{3} - 2259 T^{4} + 245 p T^{5} + 28 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2:C_4$ | \( 1 - 5 T - 19 T^{2} + 5 p T^{3} - 4 T^{4} + 5 p^{2} T^{5} - 19 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_4\times C_2$ | \( 1 - 3 T - 22 T^{2} + 159 T^{3} + 205 T^{4} + 159 p T^{5} - 22 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 - 7 T - 18 T^{2} + 145 T^{3} + 371 T^{4} + 145 p T^{5} - 18 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 - 8 T - 17 T^{2} + 254 T^{3} - 435 T^{4} + 254 p T^{5} - 17 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 3 T + 77 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 - 2 T - 43 T^{2} - 50 T^{3} + 2351 T^{4} - 50 p T^{5} - 43 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 + 9 T + 8 T^{2} + 315 T^{3} + 5131 T^{4} + 315 p T^{5} + 8 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 + 31 T^{2} + 210 T^{3} + 2851 T^{4} + 210 p T^{5} + 31 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 - 13 T + 78 T^{2} - 941 T^{3} + 11075 T^{4} - 941 p T^{5} + 78 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^2:C_4$ | \( 1 - 2 T - 3 T^{2} + 410 T^{3} + 1601 T^{4} + 410 p T^{5} - 3 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2:C_4$ | \( 1 - 8 T - 37 T^{2} + 694 T^{3} - 2425 T^{4} + 694 p T^{5} - 37 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_4\times C_2$ | \( 1 + 9 T + 8 T^{2} - 585 T^{3} - 5849 T^{4} - 585 p T^{5} + 8 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2:C_4$ | \( 1 - 15 T + 21 T^{2} + 145 T^{3} + 2916 T^{4} + 145 p T^{5} + 21 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 + 9 T - 52 T^{2} - 675 T^{3} + 121 T^{4} - 675 p T^{5} - 52 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_4\times C_2$ | \( 1 + 20 T + 151 T^{2} + 1600 T^{3} + 21441 T^{4} + 1600 p T^{5} + 151 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^2:C_4$ | \( 1 + 8 T - 63 T^{2} - 20 T^{3} + 9821 T^{4} - 20 p T^{5} - 63 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−10.06250877612428110861837420383, −9.457492555984399402444689845909, −9.344765813345424687886412713960, −9.199137839270798980881721888229, −8.827104062926329491716274929933, −8.169094411065815224186471539512, −8.064968574265870627198111806531, −7.963877643924952661272178780234, −7.53106459787801988135621123294, −7.13074336729048113073877128460, −6.94645460846324809154954564180, −6.64732879829933377158895834460, −6.62837148358234187458345868320, −5.92607240875110110145163035708, −5.37505238086176902259383955208, −5.08641636657488064733419266759, −4.70163256403741205678875710037, −4.66902844881642411729754878148, −4.41118346850708511770952688303, −4.20400976548610655520605483032, −3.55477354531954723538833763218, −2.72787109694886022922365888462, −2.59825336491970313666444998998, −2.21996412766724760751917484491, −1.58565721465164041901166539504,
1.58565721465164041901166539504, 2.21996412766724760751917484491, 2.59825336491970313666444998998, 2.72787109694886022922365888462, 3.55477354531954723538833763218, 4.20400976548610655520605483032, 4.41118346850708511770952688303, 4.66902844881642411729754878148, 4.70163256403741205678875710037, 5.08641636657488064733419266759, 5.37505238086176902259383955208, 5.92607240875110110145163035708, 6.62837148358234187458345868320, 6.64732879829933377158895834460, 6.94645460846324809154954564180, 7.13074336729048113073877128460, 7.53106459787801988135621123294, 7.963877643924952661272178780234, 8.064968574265870627198111806531, 8.169094411065815224186471539512, 8.827104062926329491716274929933, 9.199137839270798980881721888229, 9.344765813345424687886412713960, 9.457492555984399402444689845909, 10.06250877612428110861837420383
