| L(s) = 1 | − 10·4-s + 75·16-s + 800·25-s − 80·31-s − 560·37-s + 2.12e3·49-s − 1.14e3·64-s + 2.08e3·67-s + 8.80e3·97-s − 8.00e3·100-s − 7.52e3·103-s + 676·121-s + 800·124-s + 127-s + 131-s + 137-s + 139-s + 5.60e3·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.40e3·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 5/4·4-s + 1.17·16-s + 32/5·25-s − 0.463·31-s − 2.48·37-s + 6.18·49-s − 2.22·64-s + 3.79·67-s + 9.21·97-s − 8·100-s − 7.19·103-s + 0.507·121-s + 0.579·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 3.11·148-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 0.637·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.362865633\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.362865633\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - 676 T^{2} - 14154 p^{2} T^{4} - 676 p^{6} T^{6} + p^{12} T^{8} \) |
| good | 2 | \( ( 1 + 5 T^{2} + 5 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 5 | \( ( 1 - 8 p^{2} T^{2} + p^{6} T^{4} )^{4} \) |
| 7 | \( ( 1 - 1060 T^{2} + 514050 T^{4} - 1060 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 13 | \( ( 1 - 700 T^{2} + 5230950 T^{4} - 700 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 17 | \( ( 1 + 6080 T^{2} + 14812350 T^{4} + 6080 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 19 | \( ( 1 - 18436 T^{2} + 166980786 T^{4} - 18436 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 23 | \( ( 1 + 9932 T^{2} + 57602934 T^{4} + 9932 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 29 | \( ( 1 + 69056 T^{2} + 2380486926 T^{4} + 69056 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 31 | \( ( 1 + 20 T + 40350 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 37 | \( ( 1 + 140 T + 52506 T^{2} + 140 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 41 | \( ( 1 + 163184 T^{2} + 15447280446 T^{4} + 163184 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 43 | \( ( 1 - 240340 T^{2} + 26312850450 T^{4} - 240340 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 47 | \( ( 1 - 149846 T^{2} + p^{6} T^{4} )^{4} \) |
| 53 | \( ( 1 - 380608 T^{2} + 80522854674 T^{4} - 380608 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 59 | \( ( 1 - 314500 T^{2} + 97626686982 T^{4} - 314500 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 61 | \( ( 1 - 377524 T^{2} + 127308921366 T^{4} - 377524 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 67 | \( ( 1 - 260 T + p^{3} T^{2} )^{8} \) |
| 71 | \( ( 1 - 978700 T^{2} + 444510659142 T^{4} - 978700 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 73 | \( ( 1 + 379940 T^{2} + 218125296150 T^{4} + 379940 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 79 | \( ( 1 - 160756 T^{2} + 483387051426 T^{4} - 160756 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 83 | \( ( 1 + 1970780 T^{2} + 1609400316150 T^{4} + 1970780 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 89 | \( ( 1 - 1401424 T^{2} + 982134568866 T^{4} - 1401424 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 97 | \( ( 1 - 2200 T + 2552046 T^{2} - 2200 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
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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
−5.97724073314869673571793506554, −5.63108985443938994705947591344, −5.51522873156281486848838766787, −5.23965957533741317478846544450, −5.23439529866529323825736240340, −5.20967757345229288175472990744, −4.83365329074903548123375771090, −4.76181372811766746769088005796, −4.66893560433983426044055522571, −4.23558615121352822911528988264, −4.11072530015319317112594811754, −4.01892062113478652049515215829, −3.67522681921224379340901119962, −3.37046151588962614975612165024, −3.28954342758646696344039579399, −3.26974492363847643678859257167, −2.59597655020485297334633156775, −2.54353184525994104238703694397, −2.52547317147681956120262272061, −2.08385315222652905755081719439, −1.47432702258862953238167277875, −1.30008869453229184012365063684, −0.77073262116733273510402524057, −0.72795141777962244672796958517, −0.51646543360713453030962755960,
0.51646543360713453030962755960, 0.72795141777962244672796958517, 0.77073262116733273510402524057, 1.30008869453229184012365063684, 1.47432702258862953238167277875, 2.08385315222652905755081719439, 2.52547317147681956120262272061, 2.54353184525994104238703694397, 2.59597655020485297334633156775, 3.26974492363847643678859257167, 3.28954342758646696344039579399, 3.37046151588962614975612165024, 3.67522681921224379340901119962, 4.01892062113478652049515215829, 4.11072530015319317112594811754, 4.23558615121352822911528988264, 4.66893560433983426044055522571, 4.76181372811766746769088005796, 4.83365329074903548123375771090, 5.20967757345229288175472990744, 5.23439529866529323825736240340, 5.23965957533741317478846544450, 5.51522873156281486848838766787, 5.63108985443938994705947591344, 5.97724073314869673571793506554
Plot not available for L-functions of degree greater than 10.
