| L(s) = 1 | + 68·19-s − 4·41-s − 8·71-s − 28·79-s + 22·81-s + 120·89-s − 28·101-s + 24·109-s + 94·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | + 15.6·19-s − 0.624·41-s − 0.949·71-s − 3.15·79-s + 22/9·81-s + 12.7·89-s − 2.78·101-s + 2.29·109-s + 8.54·121-s + 0.0887·127-s + 0.0873·131-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 + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{32} \cdot 23^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{32} \cdot 23^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(451.2760784\) |
| \(L(\frac12)\) |
\(\approx\) |
\(451.2760784\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + 16 T^{2} + 260 T^{4} - 16080 T^{6} - 346042 T^{8} - 16080 p^{2} T^{10} + 260 p^{4} T^{12} + 16 p^{6} T^{14} + p^{8} T^{16} \) |
| good | 3 | \( 1 - 22 T^{4} + 217 T^{8} - 281 p^{2} T^{12} + 122 p^{5} T^{16} - 281 p^{6} T^{20} + 217 p^{8} T^{24} - 22 p^{12} T^{28} + p^{16} T^{32} \) |
| 7 | \( 1 - 3 T^{4} - 1606 T^{8} + 10419 T^{12} + 429738 T^{16} + 10419 p^{4} T^{20} - 1606 p^{8} T^{24} - 3 p^{12} T^{28} + p^{16} T^{32} \) |
| 11 | \( ( 1 - 47 T^{2} + 1154 T^{4} - 19401 T^{6} + 243962 T^{8} - 19401 p^{2} T^{10} + 1154 p^{4} T^{12} - 47 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 13 | \( 1 + 479 T^{4} + 134065 T^{8} + 27198575 T^{12} + 4611197425 T^{16} + 27198575 p^{4} T^{20} + 134065 p^{8} T^{24} + 479 p^{12} T^{28} + p^{16} T^{32} \) |
| 17 | \( 1 - 56 p T^{4} + 496540 T^{8} - 173841928 T^{12} + 52685394502 T^{16} - 173841928 p^{4} T^{20} + 496540 p^{8} T^{24} - 56 p^{13} T^{28} + p^{16} T^{32} \) |
| 19 | \( ( 1 - 17 T + 170 T^{2} - 1161 T^{3} + 5858 T^{4} - 1161 p T^{5} + 170 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 29 | \( ( 1 - 153 T^{2} + 10801 T^{4} - 484569 T^{6} + 15986217 T^{8} - 484569 p^{2} T^{10} + 10801 p^{4} T^{12} - 153 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 31 | \( ( 1 + 97 T^{2} - 63 T^{3} + 4056 T^{4} - 63 p T^{5} + 97 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 37 | \( 1 - 4824 T^{4} + 7557020 T^{8} + 206703000 T^{12} - 11162927645946 T^{16} + 206703000 p^{4} T^{20} + 7557020 p^{8} T^{24} - 4824 p^{12} T^{28} + p^{16} T^{32} \) |
| 41 | \( ( 1 + T + 153 T^{2} + 111 T^{3} + 9199 T^{4} + 111 p T^{5} + 153 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 43 | \( 1 - 9123 T^{4} + 42156074 T^{8} - 128578605501 T^{12} + 279586773608778 T^{16} - 128578605501 p^{4} T^{20} + 42156074 p^{8} T^{24} - 9123 p^{12} T^{28} + p^{16} T^{32} \) |
| 47 | \( 1 - 150 T^{4} + 808145 T^{8} - 2522145441 T^{12} - 38471542540050 T^{16} - 2522145441 p^{4} T^{20} + 808145 p^{8} T^{24} - 150 p^{12} T^{28} + p^{16} T^{32} \) |
| 53 | \( 1 + 6792 T^{4} + 14424284 T^{8} + 10602012216 T^{12} + 11093376541062 T^{16} + 10602012216 p^{4} T^{20} + 14424284 p^{8} T^{24} + 6792 p^{12} T^{28} + p^{16} T^{32} \) |
| 59 | \( ( 1 - 137 T^{2} + 14285 T^{4} - 912222 T^{6} + 60533186 T^{8} - 912222 p^{2} T^{10} + 14285 p^{4} T^{12} - 137 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 61 | \( ( 1 - 120 T^{2} + 9404 T^{4} - 266376 T^{6} + 10975398 T^{8} - 266376 p^{2} T^{10} + 9404 p^{4} T^{12} - 120 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 67 | \( 1 + 1800 T^{4} + 4681820 T^{8} - 36698825544 T^{12} + 304450671025926 T^{16} - 36698825544 p^{4} T^{20} + 4681820 p^{8} T^{24} + 1800 p^{12} T^{28} + p^{16} T^{32} \) |
| 71 | \( ( 1 + 2 T + 183 T^{2} + 255 T^{3} + 16144 T^{4} + 255 p T^{5} + 183 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 73 | \( 1 + 191 T^{4} + 48432625 T^{8} + 109255388975 T^{12} + 1654831419334609 T^{16} + 109255388975 p^{4} T^{20} + 48432625 p^{8} T^{24} + 191 p^{12} T^{28} + p^{16} T^{32} \) |
| 79 | \( ( 1 + 7 T + 188 T^{2} + 1143 T^{3} + 21518 T^{4} + 1143 p T^{5} + 188 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 83 | \( 1 - 26659 T^{4} + 210831370 T^{8} + 363508501475 T^{12} - 12091984145521526 T^{16} + 363508501475 p^{4} T^{20} + 210831370 p^{8} T^{24} - 26659 p^{12} T^{28} + p^{16} T^{32} \) |
| 89 | \( ( 1 - 30 T + 548 T^{2} - 7506 T^{3} + 80838 T^{4} - 7506 p T^{5} + 548 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 97 | \( 1 + 11784 T^{4} + 99996956 T^{8} - 478069607880 T^{12} - 7987409322929082 T^{16} - 478069607880 p^{4} T^{20} + 99996956 p^{8} T^{24} + 11784 p^{12} T^{28} + p^{16} T^{32} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.18661484124322368418029047891, −2.15465287026242714638376710017, −2.10640034034083557749350237405, −1.99177086697224774454267144658, −1.97557207319050846389950606874, −1.93446421257461940838020693635, −1.82063694911297631051989491009, −1.54111605681945863868466247574, −1.50744831072184790926048635296, −1.49396612553988540825206577597, −1.32845900667436637394890986479, −1.30083290007116303897179815156, −1.29322948069448412650746566230, −1.29110009987189960576557134474, −1.28109537906094329140921480363, −0.966813291718448129939258556196, −0.957090928164729227542818087319, −0.78409442122326209391772983741, −0.77188751781037577758878225895, −0.70117200797015327331874040913, −0.69368366221174641608777240416, −0.60972947295577706275085443636, −0.51930225862526430273056973593, −0.49394804787645576799027446088, −0.22143173557649833403436040943,
0.22143173557649833403436040943, 0.49394804787645576799027446088, 0.51930225862526430273056973593, 0.60972947295577706275085443636, 0.69368366221174641608777240416, 0.70117200797015327331874040913, 0.77188751781037577758878225895, 0.78409442122326209391772983741, 0.957090928164729227542818087319, 0.966813291718448129939258556196, 1.28109537906094329140921480363, 1.29110009987189960576557134474, 1.29322948069448412650746566230, 1.30083290007116303897179815156, 1.32845900667436637394890986479, 1.49396612553988540825206577597, 1.50744831072184790926048635296, 1.54111605681945863868466247574, 1.82063694911297631051989491009, 1.93446421257461940838020693635, 1.97557207319050846389950606874, 1.99177086697224774454267144658, 2.10640034034083557749350237405, 2.15465287026242714638376710017, 2.18661484124322368418029047891
Plot not available for L-functions of degree greater than 10.
