| L(s) = 1 | + 4·3-s − 8·5-s + 4·7-s + 4·9-s + 12·11-s − 32·15-s + 16·19-s + 16·21-s + 4·23-s + 30·25-s − 4·27-s − 12·29-s + 4·31-s + 48·33-s − 32·35-s − 8·37-s − 20·41-s − 16·43-s − 32·45-s + 4·49-s + 12·53-s − 96·55-s + 64·57-s + 8·59-s + 8·61-s + 16·63-s + 16·69-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 3.57·5-s + 1.51·7-s + 4/3·9-s + 3.61·11-s − 8.26·15-s + 3.67·19-s + 3.49·21-s + 0.834·23-s + 6·25-s − 0.769·27-s − 2.22·29-s + 0.718·31-s + 8.35·33-s − 5.40·35-s − 1.31·37-s − 3.12·41-s − 2.43·43-s − 4.77·45-s + 4/7·49-s + 1.64·53-s − 12.9·55-s + 8.47·57-s + 1.04·59-s + 1.02·61-s + 2.01·63-s + 1.92·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.908422871\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.908422871\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 4 T + 4 p T^{2} - 28 T^{3} + 56 T^{4} - 28 p T^{5} + 4 p^{3} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2$$\times$$C_2^2$ | \( ( 1 + 4 T + p T^{2} )^{2}( 1 + 8 T^{2} + p^{2} T^{4} ) \) |
| 7 | $D_4\times C_2$ | \( 1 - 4 T + 12 T^{2} - 44 T^{3} + 120 T^{4} - 44 p T^{5} + 12 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 12 T + 68 T^{2} - 268 T^{3} + 920 T^{4} - 268 p T^{5} + 68 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 274 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 752 T^{3} + 3634 T^{4} - 752 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 4 T + 36 T^{2} - 244 T^{3} + 1112 T^{4} - 244 p T^{5} + 36 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 12 T + 54 T^{2} + 108 T^{3} + 162 T^{4} + 108 p T^{5} + 54 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 4 T + 4 T^{2} + 4 p T^{3} - 488 T^{4} + 4 p^{2} T^{5} + 4 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 8 T + 114 T^{2} + 712 T^{3} + 5922 T^{4} + 712 p T^{5} + 114 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 20 T + 150 T^{2} + 500 T^{3} + 1250 T^{4} + 500 p T^{5} + 150 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 624 T^{3} + 3026 T^{4} + 624 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 8806 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 660 T^{3} + 6046 T^{4} - 660 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} + 264 T^{3} - 6286 T^{4} + 264 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 8 T + 18 T^{2} + 248 T^{3} - 1950 T^{4} + 248 p T^{5} + 18 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 + 126 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 + 20 T + 108 T^{2} - 1236 T^{3} - 22088 T^{4} - 1236 p T^{5} + 108 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 4 T + 102 T^{2} + 124 T^{3} + 3138 T^{4} + 124 p T^{5} + 102 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 12 T + 68 T^{2} - 540 T^{3} + 4184 T^{4} - 540 p T^{5} + 68 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} - 72 T^{3} - 8302 T^{4} - 72 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 160 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 8 T + 18 T^{2} - 392 T^{3} - 3102 T^{4} - 392 p T^{5} + 18 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−9.629695352347948756626295373758, −9.242893830145144197679802043665, −9.001935557050451030753754798881, −8.830346983208580652371693380573, −8.406293465723830743600754873592, −8.318270024138588281427088406208, −8.243175245549444512582356854906, −7.895129857534132766342874709101, −7.66076909275871127418207004193, −7.25455182193079029685852750048, −6.92950380634399347759571040245, −6.87419274007608039196708939197, −6.80536791675378667384829325341, −5.76075044310181780463299303144, −5.39445222000120628302248431406, −5.01373469425691523005371018032, −4.81675460955519326781157788721, −4.00941102580356670507514581890, −3.96574153060757612134294273856, −3.70721313816379295080794772486, −3.38106650265952903260148178650, −3.28389998045347009998040884321, −2.80719149161342445075862457227, −1.57482980005545839984243177946, −1.38053787386229656749886275517,
1.38053787386229656749886275517, 1.57482980005545839984243177946, 2.80719149161342445075862457227, 3.28389998045347009998040884321, 3.38106650265952903260148178650, 3.70721313816379295080794772486, 3.96574153060757612134294273856, 4.00941102580356670507514581890, 4.81675460955519326781157788721, 5.01373469425691523005371018032, 5.39445222000120628302248431406, 5.76075044310181780463299303144, 6.80536791675378667384829325341, 6.87419274007608039196708939197, 6.92950380634399347759571040245, 7.25455182193079029685852750048, 7.66076909275871127418207004193, 7.895129857534132766342874709101, 8.243175245549444512582356854906, 8.318270024138588281427088406208, 8.406293465723830743600754873592, 8.830346983208580652371693380573, 9.001935557050451030753754798881, 9.242893830145144197679802043665, 9.629695352347948756626295373758
