| L(s) = 1 | − 2·3-s + 6·5-s − 14·7-s + 2·9-s + 20·11-s + 32·13-s − 12·15-s + 16·17-s + 28·21-s − 38·23-s + 18·25-s + 22·27-s + 36·31-s − 40·33-s − 84·35-s + 60·37-s − 64·39-s + 52·41-s + 94·43-s + 12·45-s + 106·47-s + 98·49-s − 32·51-s + 12·53-s + 120·55-s + 204·61-s − 28·63-s + ⋯ |
| L(s) = 1 | − 2/3·3-s + 6/5·5-s − 2·7-s + 2/9·9-s + 1.81·11-s + 2.46·13-s − 4/5·15-s + 0.941·17-s + 4/3·21-s − 1.65·23-s + 0.719·25-s + 0.814·27-s + 1.16·31-s − 1.21·33-s − 2.39·35-s + 1.62·37-s − 1.64·39-s + 1.26·41-s + 2.18·43-s + 4/15·45-s + 2.25·47-s + 2·49-s − 0.627·51-s + 0.226·53-s + 2.18·55-s + 3.34·61-s − 4/9·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(5.311458014\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.311458014\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 + 2 T + 2 T^{2} - 22 T^{3} - 158 T^{4} - 22 p^{2} T^{5} + 2 p^{4} T^{6} + 2 p^{6} T^{7} + p^{8} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 2 p T + 2 p^{2} T^{2} + 106 p T^{3} + 5602 T^{4} + 106 p^{3} T^{5} + 2 p^{6} T^{6} + 2 p^{7} T^{7} + p^{8} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 10 T + 226 T^{2} - 10 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 32 T + 512 T^{2} - 6880 T^{3} + 90334 T^{4} - 6880 p^{2} T^{5} + 512 p^{4} T^{6} - 32 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 3824 T^{3} + 111742 T^{4} - 3824 p^{2} T^{5} + 128 p^{4} T^{6} - 16 p^{6} T^{7} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 916 T^{2} + 404806 T^{4} - 916 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 38 T + 722 T^{2} + 19950 T^{3} + 551234 T^{4} + 19950 p^{2} T^{5} + 722 p^{4} T^{6} + 38 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 1252 T^{2} + 756838 T^{4} - 1252 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 18 T + 1962 T^{2} - 18 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 60 T + 1800 T^{2} - 30420 T^{3} - 228946 T^{4} - 30420 p^{2} T^{5} + 1800 p^{4} T^{6} - 60 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 26 T + 210 T^{2} - 26 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 94 T + 4418 T^{2} - 275702 T^{3} + 16029922 T^{4} - 275702 p^{2} T^{5} + 4418 p^{4} T^{6} - 94 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 106 T + 5618 T^{2} - 328706 T^{3} + 18436738 T^{4} - 328706 p^{2} T^{5} + 5618 p^{4} T^{6} - 106 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 7764 T^{2} + 37391750 T^{4} - 7764 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 102 T + 10002 T^{2} - 102 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 126 T + 7938 T^{2} - 813078 T^{3} + 79425122 T^{4} - 813078 p^{2} T^{5} + 7938 p^{4} T^{6} - 126 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 78 T + 11562 T^{2} + 78 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 148 T + 10952 T^{2} - 999740 T^{3} + 89226574 T^{4} - 999740 p^{2} T^{5} + 10952 p^{4} T^{6} - 148 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 3460 T^{2} + 70145158 T^{4} - 3460 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 186 T + 17298 T^{2} + 1624338 T^{3} + 149130242 T^{4} + 1624338 p^{2} T^{5} + 17298 p^{4} T^{6} + 186 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 23236 T^{2} + 243668806 T^{4} - 23236 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 98 T + 4802 T^{2} - 98 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−8.329738540091108708269192743498, −8.023276266446862877031464142818, −7.82741714565466634444174159151, −7.19251931069984786075094205804, −7.14495154058365962876204736699, −6.70208388680182781928560582803, −6.45628232402408029514874910106, −6.45313365396360152124547869051, −6.23160001362483198439672670299, −5.79752917352181862247861420271, −5.73043739108590760397318812563, −5.55243429560831550365318748767, −5.41327514108352476704443880225, −4.39589018945226401025175591982, −4.32618137757264571064529993286, −4.16676796241867800708353046422, −3.79361425910170647820350818807, −3.50689552498725050892440708147, −3.24139143822930353650988229690, −2.63567607161585810234103780851, −2.41333923409322867352793637318, −2.00493516466914997938600194676, −1.08682453694451839273048749184, −0.973032052261883982807820710450, −0.77887677778254381869791567415,
0.77887677778254381869791567415, 0.973032052261883982807820710450, 1.08682453694451839273048749184, 2.00493516466914997938600194676, 2.41333923409322867352793637318, 2.63567607161585810234103780851, 3.24139143822930353650988229690, 3.50689552498725050892440708147, 3.79361425910170647820350818807, 4.16676796241867800708353046422, 4.32618137757264571064529993286, 4.39589018945226401025175591982, 5.41327514108352476704443880225, 5.55243429560831550365318748767, 5.73043739108590760397318812563, 5.79752917352181862247861420271, 6.23160001362483198439672670299, 6.45313365396360152124547869051, 6.45628232402408029514874910106, 6.70208388680182781928560582803, 7.14495154058365962876204736699, 7.19251931069984786075094205804, 7.82741714565466634444174159151, 8.023276266446862877031464142818, 8.329738540091108708269192743498
