| L(s) = 1 | − 2·2-s + 4-s + 34·8-s + 34·9-s − 100·11-s + 44·16-s − 68·18-s + 200·22-s − 352·23-s + 314·25-s + 520·29-s − 78·32-s + 34·36-s − 212·37-s + 1.08e3·43-s − 100·44-s + 704·46-s − 628·50-s − 16·53-s − 1.04e3·58-s + 371·64-s + 1.94e3·67-s + 4.49e3·71-s + 1.15e3·72-s + 424·74-s + 1.04e3·79-s + 1.16e3·81-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/8·4-s + 1.50·8-s + 1.25·9-s − 2.74·11-s + 0.687·16-s − 0.890·18-s + 1.93·22-s − 3.19·23-s + 2.51·25-s + 3.32·29-s − 0.430·32-s + 0.157·36-s − 0.941·37-s + 3.83·43-s − 0.342·44-s + 2.25·46-s − 1.77·50-s − 0.0414·53-s − 2.35·58-s + 0.724·64-s + 3.54·67-s + 7.51·71-s + 1.89·72-s + 0.666·74-s + 1.49·79-s + 1.59·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.590715460\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.590715460\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( ( 1 + T + T^{2} - p^{4} T^{3} - 9 p^{3} T^{4} - p^{7} T^{5} + p^{6} T^{6} + p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 3 | \( 1 - 34 T^{2} - 2 p T^{4} + 10064 T^{6} - 23921 T^{8} + 10064 p^{6} T^{10} - 2 p^{13} T^{12} - 34 p^{18} T^{14} + p^{24} T^{16} \) |
| 5 | \( 1 - 314 T^{2} + 50562 T^{4} - 5270176 T^{6} + 522562031 T^{8} - 5270176 p^{6} T^{10} + 50562 p^{12} T^{12} - 314 p^{18} T^{14} + p^{24} T^{16} \) |
| 11 | \( ( 1 + 50 T - 202 T^{2} + 2000 T^{3} + 2201743 T^{4} + 2000 p^{3} T^{5} - 202 p^{6} T^{6} + 50 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 13 | \( ( 1 + 5554 T^{2} + 17209282 T^{4} + 5554 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 17 | \( 1 - 5088 T^{2} + 14421310 T^{4} + 187282685952 T^{6} - 1076010335607741 T^{8} + 187282685952 p^{6} T^{10} + 14421310 p^{12} T^{12} - 5088 p^{18} T^{14} + p^{24} T^{16} \) |
| 19 | \( 1 - 12690 T^{2} + 28454938 T^{4} - 488430486000 T^{6} + 7788364400899983 T^{8} - 488430486000 p^{6} T^{10} + 28454938 p^{12} T^{12} - 12690 p^{18} T^{14} + p^{24} T^{16} \) |
| 23 | \( ( 1 + 176 T - 62 T^{2} + 1179904 T^{3} + 438436563 T^{4} + 1179904 p^{3} T^{5} - 62 p^{6} T^{6} + 176 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 29 | \( ( 1 - 130 T + 49818 T^{2} - 130 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 31 | \( 1 - 19060 T^{2} + 373250298 T^{4} + 34021605583600 T^{6} - 1109978980966458157 T^{8} + 34021605583600 p^{6} T^{10} + 373250298 p^{12} T^{12} - 19060 p^{18} T^{14} + p^{24} T^{16} \) |
| 37 | \( ( 1 + 106 T - 92294 T^{2} + 235744 T^{3} + 7583597383 T^{4} + 235744 p^{3} T^{5} - 92294 p^{6} T^{6} + 106 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 41 | \( ( 1 + 208848 T^{2} + 19595539298 T^{4} + 208848 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 43 | \( ( 1 - 270 T + 97614 T^{2} - 270 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 47 | \( 1 - 228052 T^{2} + 17880021370 T^{4} - 2866445491787152 T^{6} + \)\(48\!\cdots\!99\)\( T^{8} - 2866445491787152 p^{6} T^{10} + 17880021370 p^{12} T^{12} - 228052 p^{18} T^{14} + p^{24} T^{16} \) |
| 53 | \( ( 1 + 8 T - 276646 T^{2} - 168352 T^{3} + 54394534843 T^{4} - 168352 p^{3} T^{5} - 276646 p^{6} T^{6} + 8 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 59 | \( 1 - 632162 T^{2} + 222723302586 T^{4} - 58503068402380912 T^{6} + \)\(12\!\cdots\!79\)\( T^{8} - 58503068402380912 p^{6} T^{10} + 222723302586 p^{12} T^{12} - 632162 p^{18} T^{14} + p^{24} T^{16} \) |
| 61 | \( 1 - 8978 p T^{2} + 149331839266 T^{4} - 426964025450528 p T^{6} + \)\(45\!\cdots\!79\)\( T^{8} - 426964025450528 p^{7} T^{10} + 149331839266 p^{12} T^{12} - 8978 p^{19} T^{14} + p^{24} T^{16} \) |
| 67 | \( ( 1 - 972 T + 200922 T^{2} - 138350592 T^{3} + 178716222683 T^{4} - 138350592 p^{3} T^{5} + 200922 p^{6} T^{6} - 972 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 71 | \( ( 1 - 1124 T + 1018926 T^{2} - 1124 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 73 | \( 1 - 1279312 T^{2} + 931531176990 T^{4} - 514845763213402112 T^{6} + \)\(22\!\cdots\!79\)\( T^{8} - 514845763213402112 p^{6} T^{10} + 931531176990 p^{12} T^{12} - 1279312 p^{18} T^{14} + p^{24} T^{16} \) |
| 79 | \( ( 1 - 524 T - 205806 T^{2} + 264984704 T^{3} - 147697266061 T^{4} + 264984704 p^{3} T^{5} - 205806 p^{6} T^{6} - 524 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 83 | \( ( 1 + 2275682 T^{2} + 1948536513034 T^{4} + 2275682 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 89 | \( 1 + 804000 T^{2} - 70725036962 T^{4} - 222564522147840000 T^{6} - \)\(53\!\cdots\!77\)\( T^{8} - 222564522147840000 p^{6} T^{10} - 70725036962 p^{12} T^{12} + 804000 p^{18} T^{14} + p^{24} T^{16} \) |
| 97 | \( ( 1 + 567808 T^{2} + 211724872834 T^{4} + 567808 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
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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
−7.24184998822917238753037461662, −6.83338883670394509606544857391, −6.67424031866734028450471019162, −6.34950089040177703848099872359, −6.33880726633345803034014811769, −6.17096336750037277830043142027, −5.95731012561353946710290128700, −5.37276842199620785742187338700, −5.21526535009488464766798305784, −5.15227549073786800362189762183, −5.14311828032920619167830366237, −4.86072411061712537847113237302, −4.62711298756083186187153665339, −4.20923120209997597634682106564, −3.90370963521978325696160293555, −3.85175445141334880018169964442, −3.74036059967678764411312221647, −3.14884325327222913647353029236, −2.69872940460800571062035467619, −2.52407208301227235143754236447, −2.25176300575461817891705979816, −2.07779906582429401816152577618, −1.18779700036515412361999396350, −0.873325273054187424694295672963, −0.71187396613198385835126615551,
0.71187396613198385835126615551, 0.873325273054187424694295672963, 1.18779700036515412361999396350, 2.07779906582429401816152577618, 2.25176300575461817891705979816, 2.52407208301227235143754236447, 2.69872940460800571062035467619, 3.14884325327222913647353029236, 3.74036059967678764411312221647, 3.85175445141334880018169964442, 3.90370963521978325696160293555, 4.20923120209997597634682106564, 4.62711298756083186187153665339, 4.86072411061712537847113237302, 5.14311828032920619167830366237, 5.15227549073786800362189762183, 5.21526535009488464766798305784, 5.37276842199620785742187338700, 5.95731012561353946710290128700, 6.17096336750037277830043142027, 6.33880726633345803034014811769, 6.34950089040177703848099872359, 6.67424031866734028450471019162, 6.83338883670394509606544857391, 7.24184998822917238753037461662
Plot not available for L-functions of degree greater than 10.
