| L(s) = 1 | + 8·7-s + 4·13-s − 8·19-s + 8·25-s + 56·31-s − 8·37-s − 4·43-s + 4·49-s − 20·61-s − 28·67-s − 20·73-s − 28·79-s + 32·91-s − 32·97-s + 8·103-s + 16·109-s − 16·121-s + 127-s + 131-s − 64·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 3.02·7-s + 1.10·13-s − 1.83·19-s + 8/5·25-s + 10.0·31-s − 1.31·37-s − 0.609·43-s + 4/7·49-s − 2.56·61-s − 3.42·67-s − 2.34·73-s − 3.15·79-s + 3.35·91-s − 3.24·97-s + 0.788·103-s + 1.53·109-s − 1.45·121-s + 0.0887·127-s + 0.0873·131-s − 5.54·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.01299394913\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01299394913\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| good | 5 | \( 1 - 8 T^{2} + 22 T^{4} + 64 T^{6} - 461 T^{8} + 64 p^{2} T^{10} + 22 p^{4} T^{12} - 8 p^{6} T^{14} + p^{8} T^{16} \) |
| 7 | \( ( 1 - 2 T + 9 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 + 8 T^{2} + 42 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} )^{4} \) |
| 17 | \( 1 - 20 T^{2} + 106 T^{4} + 5680 T^{6} - 112685 T^{8} + 5680 p^{2} T^{10} + 106 p^{4} T^{12} - 20 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( 1 - 56 T^{2} + 1318 T^{4} - 42560 T^{6} + 1366339 T^{8} - 42560 p^{2} T^{10} + 1318 p^{4} T^{12} - 56 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( 1 + 28 T^{2} - 710 T^{4} - 5264 T^{6} + 865411 T^{8} - 5264 p^{2} T^{10} - 710 p^{4} T^{12} + 28 p^{6} T^{14} + p^{8} T^{16} \) |
| 31 | \( ( 1 - 14 T + 105 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + 2 T + 51 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( 1 - 68 T^{2} + 1642 T^{4} + 25840 T^{6} - 1706381 T^{8} + 25840 p^{2} T^{10} + 1642 p^{4} T^{12} - 68 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( ( 1 + 2 T - 29 T^{2} - 106 T^{3} - 932 T^{4} - 106 p T^{5} - 29 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 - 92 T^{2} + 3466 T^{4} - 53360 T^{6} + 1403347 T^{8} - 53360 p^{2} T^{10} + 3466 p^{4} T^{12} - 92 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( 1 - 200 T^{2} + 24406 T^{4} - 1995200 T^{6} + 122922355 T^{8} - 1995200 p^{2} T^{10} + 24406 p^{4} T^{12} - 200 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( 1 - 104 T^{2} + 1174 T^{4} - 278720 T^{6} + 43200307 T^{8} - 278720 p^{2} T^{10} + 1174 p^{4} T^{12} - 104 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( ( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 + 14 T + p T^{2} )^{4}( 1 - 14 T + 129 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | \( 1 - 140 T^{2} + 5002 T^{4} - 632240 T^{6} + 88886323 T^{8} - 632240 p^{2} T^{10} + 5002 p^{4} T^{12} - 140 p^{6} T^{14} + p^{8} T^{16} \) |
| 73 | \( ( 1 + 5 T - 48 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + 14 T + 43 T^{2} - 70 T^{3} + 1684 T^{4} - 70 p T^{5} + 43 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 188 T^{2} + 19158 T^{4} + 188 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( 1 - 56 T^{2} + 1510 T^{4} + 796096 T^{6} - 84938621 T^{8} + 796096 p^{2} T^{10} + 1510 p^{4} T^{12} - 56 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( ( 1 + 16 T + 22 T^{2} + 640 T^{3} + 20515 T^{4} + 640 p T^{5} + 22 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−3.68554133986650274466078134014, −3.31568531549487497121674861775, −3.29821274997912090051984989706, −3.16544232375494819346943649335, −3.12977870582356840480318497549, −3.07076986235672709601776021540, −2.93026103531456084452607152311, −2.85495220483635396581231722797, −2.82275564538882994472971357899, −2.77259430325079765155271929683, −2.37748991150016003630511826696, −2.28905909572923242339982342364, −2.09110111613293957116582816196, −2.00218844831296512510144478727, −1.87752698842807773057003658851, −1.76472780089745562829090304961, −1.52343312024420248087002005180, −1.37474455466796349619980555794, −1.37159603277123062616988382934, −1.17137372113215005835756779457, −0.978506460604036322407848941853, −0.939313067009083076818473614611, −0.899301256752125751652623653605, −0.35244167096086664316184137844, −0.00685250573126656047620545199,
0.00685250573126656047620545199, 0.35244167096086664316184137844, 0.899301256752125751652623653605, 0.939313067009083076818473614611, 0.978506460604036322407848941853, 1.17137372113215005835756779457, 1.37159603277123062616988382934, 1.37474455466796349619980555794, 1.52343312024420248087002005180, 1.76472780089745562829090304961, 1.87752698842807773057003658851, 2.00218844831296512510144478727, 2.09110111613293957116582816196, 2.28905909572923242339982342364, 2.37748991150016003630511826696, 2.77259430325079765155271929683, 2.82275564538882994472971357899, 2.85495220483635396581231722797, 2.93026103531456084452607152311, 3.07076986235672709601776021540, 3.12977870582356840480318497549, 3.16544232375494819346943649335, 3.29821274997912090051984989706, 3.31568531549487497121674861775, 3.68554133986650274466078134014
Plot not available for L-functions of degree greater than 10.
