| L(s) = 1 | − 4·4-s + 12·5-s + 4·16-s + 24·17-s − 24·19-s − 48·20-s + 60·23-s + 78·25-s + 68·31-s − 240·47-s − 8·49-s − 408·53-s + 196·61-s + 16·64-s − 96·68-s + 96·76-s − 180·79-s + 48·80-s + 108·83-s + 288·85-s − 240·92-s − 288·95-s − 312·100-s + 288·107-s − 152·109-s + 48·113-s + 720·115-s + ⋯ |
| L(s) = 1 | − 4-s + 12/5·5-s + 1/4·16-s + 1.41·17-s − 1.26·19-s − 2.39·20-s + 2.60·23-s + 3.11·25-s + 2.19·31-s − 5.10·47-s − 0.163·49-s − 7.69·53-s + 3.21·61-s + 1/4·64-s − 1.41·68-s + 1.26·76-s − 2.27·79-s + 3/5·80-s + 1.30·83-s + 3.38·85-s − 2.60·92-s − 3.03·95-s − 3.11·100-s + 2.69·107-s − 1.39·109-s + 0.424·113-s + 6.26·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(11.73401580\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.73401580\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 12 T + 66 T^{2} - 336 T^{3} + 1859 T^{4} - 336 p^{2} T^{5} + 66 p^{4} T^{6} - 12 p^{6} T^{7} + p^{8} T^{8} \) |
| good | 7 | \( 1 + 8 T^{2} + 3958 T^{4} - 69568 T^{6} + 9244771 T^{8} - 69568 p^{4} T^{10} + 3958 p^{8} T^{12} + 8 p^{12} T^{14} + p^{16} T^{16} \) |
| 11 | \( 1 + 96 T^{2} - 170 p^{2} T^{4} + 48384 T^{6} + 537915459 T^{8} + 48384 p^{4} T^{10} - 170 p^{10} T^{12} + 96 p^{12} T^{14} + p^{16} T^{16} \) |
| 13 | \( 1 + 352 T^{2} + 52006 T^{4} + 5201152 T^{6} + 814769539 T^{8} + 5201152 p^{4} T^{10} + 52006 p^{8} T^{12} + 352 p^{12} T^{14} + p^{16} T^{16} \) |
| 17 | \( ( 1 - 6 T + 489 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 19 | \( ( 1 + 6 T + 713 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 23 | \( ( 1 - 30 T + 9 T^{2} + 5010 T^{3} + 37940 T^{4} + 5010 p^{2} T^{5} + 9 p^{4} T^{6} - 30 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 29 | \( 1 + 1096 T^{2} - 41258 T^{4} - 188608448 T^{6} + 193897974019 T^{8} - 188608448 p^{4} T^{10} - 41258 p^{8} T^{12} + 1096 p^{12} T^{14} + p^{16} T^{16} \) |
| 31 | \( ( 1 - 34 T - 605 T^{2} + 5474 T^{3} + 1066684 T^{4} + 5474 p^{2} T^{5} - 605 p^{4} T^{6} - 34 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 - 68 T^{2} - 1268634 T^{4} - 68 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 41 | \( 1 + 5352 T^{2} + 16236406 T^{4} + 36157983552 T^{6} + 65020254579939 T^{8} + 36157983552 p^{4} T^{10} + 16236406 p^{8} T^{12} + 5352 p^{12} T^{14} + p^{16} T^{16} \) |
| 43 | \( 1 + 5672 T^{2} + 17968534 T^{4} + 41776821056 T^{6} + 80050185474115 T^{8} + 41776821056 p^{4} T^{10} + 17968534 p^{8} T^{12} + 5672 p^{12} T^{14} + p^{16} T^{16} \) |
| 47 | \( ( 1 + 120 T + 6774 T^{2} + 384960 T^{3} + 21466595 T^{4} + 384960 p^{2} T^{5} + 6774 p^{4} T^{6} + 120 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 + 102 T + 8057 T^{2} + 102 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 59 | \( 1 + 10848 T^{2} + 64062694 T^{4} + 318732551424 T^{6} + 1301928798359235 T^{8} + 318732551424 p^{4} T^{10} + 64062694 p^{8} T^{12} + 10848 p^{12} T^{14} + p^{16} T^{16} \) |
| 61 | \( ( 1 - 98 T + 913 T^{2} - 122402 T^{3} + 25951156 T^{4} - 122402 p^{2} T^{5} + 913 p^{4} T^{6} - 98 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( 1 - 6508 T^{2} + 10043914 T^{4} + 52012534736 T^{6} - 215603753039885 T^{8} + 52012534736 p^{4} T^{10} + 10043914 p^{8} T^{12} - 6508 p^{12} T^{14} + p^{16} T^{16} \) |
| 71 | \( ( 1 - 4288 T^{2} + 45932730 T^{4} - 4288 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 - 13280 T^{2} + 84777594 T^{4} - 13280 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 79 | \( ( 1 + 90 T + 1531 T^{2} - 532170 T^{3} - 46350420 T^{4} - 532170 p^{2} T^{5} + 1531 p^{4} T^{6} + 90 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 83 | \( ( 1 - 54 T - 4863 T^{2} + 323946 T^{3} - 7033804 T^{4} + 323946 p^{2} T^{5} - 4863 p^{4} T^{6} - 54 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 - 8136 T^{2} + 91108874 T^{4} - 8136 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 97 | \( 1 + 19172 T^{2} + 99652426 T^{4} + 1741864314512 T^{6} + 31399709763290899 T^{8} + 1741864314512 p^{4} T^{10} + 99652426 p^{8} T^{12} + 19172 p^{12} T^{14} + p^{16} T^{16} \) |
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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
−4.20244624814411789736612727171, −4.17320856238424458185769385866, −4.01586599704081018318959848927, −3.65863208361355904356344094089, −3.44681203753931436365592326374, −3.37491418734810926249250947844, −3.33064447244414224376842587832, −3.29212662110728344053144128351, −3.17069256779262911946186760453, −3.01879844238233880076745554032, −2.77546701847063028294181370914, −2.56273271193439686041251669605, −2.44843960571936123371808196261, −2.38410832108098946463886595090, −2.15185148449617643670694841433, −2.01197622597557313278541947674, −1.53104509164620896448626255768, −1.50623189031948838451985269873, −1.41637144894735404375619213629, −1.40909730026339546301963139277, −1.39455570740976044690886940287, −0.843887307048187810425267786006, −0.50378024964345190638090622435, −0.44173340893598431959001338966, −0.28197362210105099892875692597,
0.28197362210105099892875692597, 0.44173340893598431959001338966, 0.50378024964345190638090622435, 0.843887307048187810425267786006, 1.39455570740976044690886940287, 1.40909730026339546301963139277, 1.41637144894735404375619213629, 1.50623189031948838451985269873, 1.53104509164620896448626255768, 2.01197622597557313278541947674, 2.15185148449617643670694841433, 2.38410832108098946463886595090, 2.44843960571936123371808196261, 2.56273271193439686041251669605, 2.77546701847063028294181370914, 3.01879844238233880076745554032, 3.17069256779262911946186760453, 3.29212662110728344053144128351, 3.33064447244414224376842587832, 3.37491418734810926249250947844, 3.44681203753931436365592326374, 3.65863208361355904356344094089, 4.01586599704081018318959848927, 4.17320856238424458185769385866, 4.20244624814411789736612727171
Plot not available for L-functions of degree greater than 10.
