| L(s) = 1 | + 16·9-s − 32·13-s − 16·17-s − 48·25-s + 80·29-s + 176·37-s + 144·41-s − 28·49-s − 48·53-s + 192·61-s + 272·73-s + 164·81-s − 80·89-s + 528·97-s + 128·101-s − 208·109-s − 160·113-s − 512·117-s + 296·121-s − 448·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 256·153-s + ⋯ |
| L(s) = 1 | + 16/9·9-s − 2.46·13-s − 0.941·17-s − 1.91·25-s + 2.75·29-s + 4.75·37-s + 3.51·41-s − 4/7·49-s − 0.905·53-s + 3.14·61-s + 3.72·73-s + 2.02·81-s − 0.898·89-s + 5.44·97-s + 1.26·101-s − 1.90·109-s − 1.41·113-s − 4.37·117-s + 2.44·121-s − 3.58·125-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 − 1.67·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(12.08218657\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.08218657\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( ( 1 + p T^{2} )^{4} \) |
| good | 3 | \( 1 - 16 T^{2} + 92 T^{4} + 272 T^{6} - 8570 T^{8} + 272 p^{4} T^{10} + 92 p^{8} T^{12} - 16 p^{12} T^{14} + p^{16} T^{16} \) |
| 5 | \( ( 1 + 24 T^{2} + 224 T^{3} + 78 T^{4} + 224 p^{2} T^{5} + 24 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 11 | \( 1 - 296 T^{2} + 72860 T^{4} - 12907416 T^{6} + 1704788486 T^{8} - 12907416 p^{4} T^{10} + 72860 p^{8} T^{12} - 296 p^{12} T^{14} + p^{16} T^{16} \) |
| 13 | \( ( 1 + 16 T + 696 T^{2} + 7536 T^{3} + 176398 T^{4} + 7536 p^{2} T^{5} + 696 p^{4} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 + 8 T + 428 T^{2} - 1416 T^{3} + 75814 T^{4} - 1416 p^{2} T^{5} + 428 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 19 | \( 1 - 1936 T^{2} + 1795420 T^{4} - 1065665392 T^{6} + 449725089670 T^{8} - 1065665392 p^{4} T^{10} + 1795420 p^{8} T^{12} - 1936 p^{12} T^{14} + p^{16} T^{16} \) |
| 23 | \( 1 - 1928 T^{2} + 2366876 T^{4} - 1917103032 T^{6} + 1186521008582 T^{8} - 1917103032 p^{4} T^{10} + 2366876 p^{8} T^{12} - 1928 p^{12} T^{14} + p^{16} T^{16} \) |
| 29 | \( ( 1 - 40 T + 3660 T^{2} - 100632 T^{3} + 4741126 T^{4} - 100632 p^{2} T^{5} + 3660 p^{4} T^{6} - 40 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 31 | \( 1 - 3624 T^{2} + 5953500 T^{4} - 5905968152 T^{6} + 5382616134 p^{2} T^{8} - 5905968152 p^{4} T^{10} + 5953500 p^{8} T^{12} - 3624 p^{12} T^{14} + p^{16} T^{16} \) |
| 37 | \( ( 1 - 88 T + 3692 T^{2} - 66024 T^{3} + 510982 T^{4} - 66024 p^{2} T^{5} + 3692 p^{4} T^{6} - 88 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 41 | \( ( 1 - 72 T + 4364 T^{2} - 180408 T^{3} + 8880422 T^{4} - 180408 p^{2} T^{5} + 4364 p^{4} T^{6} - 72 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 43 | \( 1 - 4392 T^{2} + 15655580 T^{4} - 32786290584 T^{6} + 70340912004102 T^{8} - 32786290584 p^{4} T^{10} + 15655580 p^{8} T^{12} - 4392 p^{12} T^{14} + p^{16} T^{16} \) |
| 47 | \( 1 - 6696 T^{2} + 34041564 T^{4} - 108976927256 T^{6} + 286725887177670 T^{8} - 108976927256 p^{4} T^{10} + 34041564 p^{8} T^{12} - 6696 p^{12} T^{14} + p^{16} T^{16} \) |
| 53 | \( ( 1 + 24 T + 6108 T^{2} + 242920 T^{3} + 18930726 T^{4} + 242920 p^{2} T^{5} + 6108 p^{4} T^{6} + 24 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 59 | \( 1 - 21392 T^{2} + 213726428 T^{4} - 1316320178544 T^{6} + 5495654048256518 T^{8} - 1316320178544 p^{4} T^{10} + 213726428 p^{8} T^{12} - 21392 p^{12} T^{14} + p^{16} T^{16} \) |
| 61 | \( ( 1 - 96 T + 12728 T^{2} - 895872 T^{3} + 70065870 T^{4} - 895872 p^{2} T^{5} + 12728 p^{4} T^{6} - 96 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( 1 - 16200 T^{2} + 120433884 T^{4} - 596093100152 T^{6} + 2634880977729030 T^{8} - 596093100152 p^{4} T^{10} + 120433884 p^{8} T^{12} - 16200 p^{12} T^{14} + p^{16} T^{16} \) |
| 71 | \( 1 - 21000 T^{2} + 247457180 T^{4} - 1991232124728 T^{6} + 11615880585488070 T^{8} - 1991232124728 p^{4} T^{10} + 247457180 p^{8} T^{12} - 21000 p^{12} T^{14} + p^{16} T^{16} \) |
| 73 | \( ( 1 - 136 T + 21660 T^{2} - 1811512 T^{3} + 170528390 T^{4} - 1811512 p^{2} T^{5} + 21660 p^{4} T^{6} - 136 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 79 | \( 1 - 26248 T^{2} + 289540636 T^{4} - 1851543245752 T^{6} + 10296312398061382 T^{8} - 1851543245752 p^{4} T^{10} + 289540636 p^{8} T^{12} - 26248 p^{12} T^{14} + p^{16} T^{16} \) |
| 83 | \( 1 - 19280 T^{2} + 168996572 T^{4} - 1078781742000 T^{6} + 6945567152070790 T^{8} - 1078781742000 p^{4} T^{10} + 168996572 p^{8} T^{12} - 19280 p^{12} T^{14} + p^{16} T^{16} \) |
| 89 | \( ( 1 + 40 T + 25916 T^{2} + 837912 T^{3} + 285914566 T^{4} + 837912 p^{2} T^{5} + 25916 p^{4} T^{6} + 40 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 - 264 T + 52364 T^{2} - 7260024 T^{3} + 808802534 T^{4} - 7260024 p^{2} T^{5} + 52364 p^{4} T^{6} - 264 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
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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
−4.65713578082220318098334902861, −4.64866518115680234741300589514, −4.43566595374380455219534884013, −4.14580575428081526446981353702, −4.08652792833662535520904490072, −3.93661784930967351101725245724, −3.89419746069204789473863931089, −3.76595839737794330417719030503, −3.50659355779609984779517548744, −3.35218814684648177528320675853, −3.07224379908793761701139431391, −2.69133443278862507614580771475, −2.68913707392263347276430155669, −2.66823486594741691264045967289, −2.45993811821797887392246656966, −2.31399975609983322602482682963, −2.09906353488987261176171203065, −1.99148467226165758722971871373, −1.89346844064763723627466229157, −1.30575056083804086538103561115, −1.18956589516220750860282897463, −0.954609677208884970357914994849, −0.68621327315187088471860353336, −0.58198020875608273440697108886, −0.36600560851429686967204483019,
0.36600560851429686967204483019, 0.58198020875608273440697108886, 0.68621327315187088471860353336, 0.954609677208884970357914994849, 1.18956589516220750860282897463, 1.30575056083804086538103561115, 1.89346844064763723627466229157, 1.99148467226165758722971871373, 2.09906353488987261176171203065, 2.31399975609983322602482682963, 2.45993811821797887392246656966, 2.66823486594741691264045967289, 2.68913707392263347276430155669, 2.69133443278862507614580771475, 3.07224379908793761701139431391, 3.35218814684648177528320675853, 3.50659355779609984779517548744, 3.76595839737794330417719030503, 3.89419746069204789473863931089, 3.93661784930967351101725245724, 4.08652792833662535520904490072, 4.14580575428081526446981353702, 4.43566595374380455219534884013, 4.64866518115680234741300589514, 4.65713578082220318098334902861
Plot not available for L-functions of degree greater than 10.
