| L(s) = 1 | − 16·7-s + 12·9-s − 16·23-s + 96·29-s + 112·41-s − 128·43-s − 272·47-s + 24·49-s + 80·61-s − 192·63-s + 320·67-s + 90·81-s − 320·83-s + 48·89-s − 416·101-s + 208·103-s + 160·107-s − 112·109-s + 104·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 256·161-s + ⋯ |
| L(s) = 1 | − 2.28·7-s + 4/3·9-s − 0.695·23-s + 3.31·29-s + 2.73·41-s − 2.97·43-s − 5.78·47-s + 0.489·49-s + 1.31·61-s − 3.04·63-s + 4.77·67-s + 10/9·81-s − 3.85·83-s + 0.539·89-s − 4.11·101-s + 2.01·103-s + 1.49·107-s − 1.02·109-s + 0.859·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 1.59·161-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9916127305\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9916127305\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 - p T^{2} )^{4} \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 + 8 T + 12 p T^{2} + 664 T^{3} + 3686 T^{4} + 664 p^{2} T^{5} + 12 p^{5} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 11 | \( 1 - 104 T^{2} + 25116 T^{4} + 998312 T^{6} + 87392006 T^{8} + 998312 p^{4} T^{10} + 25116 p^{8} T^{12} - 104 p^{12} T^{14} + p^{16} T^{16} \) |
| 13 | \( 1 - 56 p T^{2} + 233244 T^{4} - 47327080 T^{6} + 8069592710 T^{8} - 47327080 p^{4} T^{10} + 233244 p^{8} T^{12} - 56 p^{13} T^{14} + p^{16} T^{16} \) |
| 17 | \( 1 + 232 T^{2} - 1956 T^{4} + 5116760 T^{6} + 4727432390 T^{8} + 5116760 p^{4} T^{10} - 1956 p^{8} T^{12} + 232 p^{12} T^{14} + p^{16} T^{16} \) |
| 19 | \( 1 - 200 T^{2} + 273628 T^{4} - 52427000 T^{6} + 42482298118 T^{8} - 52427000 p^{4} T^{10} + 273628 p^{8} T^{12} - 200 p^{12} T^{14} + p^{16} T^{16} \) |
| 23 | \( ( 1 + 8 T + 804 T^{2} + 16024 T^{3} + 464006 T^{4} + 16024 p^{2} T^{5} + 804 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 29 | \( ( 1 - 24 T + 1646 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 31 | \( 1 - 2824 T^{2} + 6208540 T^{4} - 8647788856 T^{6} + 9849609291334 T^{8} - 8647788856 p^{4} T^{10} + 6208540 p^{8} T^{12} - 2824 p^{12} T^{14} + p^{16} T^{16} \) |
| 37 | \( 1 - 4760 T^{2} + 13775388 T^{4} - 28332616360 T^{6} + 44075826632198 T^{8} - 28332616360 p^{4} T^{10} + 13775388 p^{8} T^{12} - 4760 p^{12} T^{14} + p^{16} T^{16} \) |
| 41 | \( ( 1 - 56 T + 2556 T^{2} - 74632 T^{3} + 3542726 T^{4} - 74632 p^{2} T^{5} + 2556 p^{4} T^{6} - 56 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 43 | \( ( 1 + 64 T + 7428 T^{2} + 323264 T^{3} + 20324198 T^{4} + 323264 p^{2} T^{5} + 7428 p^{4} T^{6} + 64 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 47 | \( ( 1 + 136 T + 14692 T^{2} + 970648 T^{3} + 54708550 T^{4} + 970648 p^{2} T^{5} + 14692 p^{4} T^{6} + 136 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 53 | \( 1 - 13496 T^{2} + 84948060 T^{4} - 346530467464 T^{6} + 1080061437995654 T^{8} - 346530467464 p^{4} T^{10} + 84948060 p^{8} T^{12} - 13496 p^{12} T^{14} + p^{16} T^{16} \) |
| 59 | \( 1 - 20456 T^{2} + 200671260 T^{4} - 1224887792344 T^{6} + 1465211341814 p^{2} T^{8} - 1224887792344 p^{4} T^{10} + 200671260 p^{8} T^{12} - 20456 p^{12} T^{14} + p^{16} T^{16} \) |
| 61 | \( ( 1 - 40 T + 11964 T^{2} - 456920 T^{3} + 61872806 T^{4} - 456920 p^{2} T^{5} + 11964 p^{4} T^{6} - 40 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 - 160 T + 15172 T^{2} - 1420000 T^{3} + 112390438 T^{4} - 1420000 p^{2} T^{5} + 15172 p^{4} T^{6} - 160 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 71 | \( 1 - 19720 T^{2} + 218875804 T^{4} - 1700648520760 T^{6} + 9880226933129926 T^{8} - 1700648520760 p^{4} T^{10} + 218875804 p^{8} T^{12} - 19720 p^{12} T^{14} + p^{16} T^{16} \) |
| 73 | \( 1 - 19960 T^{2} + 180624028 T^{4} - 999582302920 T^{6} + 4905138178520518 T^{8} - 999582302920 p^{4} T^{10} + 180624028 p^{8} T^{12} - 19960 p^{12} T^{14} + p^{16} T^{16} \) |
| 79 | \( 1 - 36104 T^{2} + 626466076 T^{4} - 6818196448568 T^{6} + 50837338178201926 T^{8} - 6818196448568 p^{4} T^{10} + 626466076 p^{8} T^{12} - 36104 p^{12} T^{14} + p^{16} T^{16} \) |
| 83 | \( ( 1 + 160 T + 27460 T^{2} + 2510560 T^{3} + 258587302 T^{4} + 2510560 p^{2} T^{5} + 27460 p^{4} T^{6} + 160 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 - 24 T + 4060 T^{2} + 454104 T^{3} - 10096506 T^{4} + 454104 p^{2} T^{5} + 4060 p^{4} T^{6} - 24 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 - 22780 T^{2} + 253264134 T^{4} - 22780 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.50117848004693460934606565975, −3.42518511708810385224405189219, −3.25938704210968367615613300083, −3.24852927661221417931700819047, −3.04617479366802354680585518341, −2.99528595328205932475364485388, −2.84985687862383873355068712656, −2.67563804449665221521437616757, −2.64968017797263151291606271923, −2.43829640448026534961107242768, −2.27756853085265297450370370896, −2.24827665441971148714230109376, −1.94036302130899007144222558487, −1.86557996963996597330600198676, −1.74263088565821304289196429198, −1.58489501938547193512413812241, −1.38309425096846165632277386583, −1.38281811058201971721629834141, −1.16084292835156582555899819195, −0.973106310554616974821565687049, −0.62204787484177161673230608718, −0.61316036458328036641619448799, −0.56685669391695429585542578607, −0.28558999405076316117513058632, −0.07020187617657347947774282281,
0.07020187617657347947774282281, 0.28558999405076316117513058632, 0.56685669391695429585542578607, 0.61316036458328036641619448799, 0.62204787484177161673230608718, 0.973106310554616974821565687049, 1.16084292835156582555899819195, 1.38281811058201971721629834141, 1.38309425096846165632277386583, 1.58489501938547193512413812241, 1.74263088565821304289196429198, 1.86557996963996597330600198676, 1.94036302130899007144222558487, 2.24827665441971148714230109376, 2.27756853085265297450370370896, 2.43829640448026534961107242768, 2.64968017797263151291606271923, 2.67563804449665221521437616757, 2.84985687862383873355068712656, 2.99528595328205932475364485388, 3.04617479366802354680585518341, 3.24852927661221417931700819047, 3.25938704210968367615613300083, 3.42518511708810385224405189219, 3.50117848004693460934606565975
Plot not available for L-functions of degree greater than 10.
