| L(s) = 1 | + 6·3-s + 13·9-s + 24·25-s + 8·27-s + 40·31-s + 36·37-s + 42·49-s + 44·67-s + 144·75-s − 26·81-s + 240·93-s + 64·97-s + 116·103-s + 216·111-s + 127-s + 131-s + 137-s + 139-s + 252·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 98·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 3.46·3-s + 13/3·9-s + 24/5·25-s + 1.53·27-s + 7.18·31-s + 5.91·37-s + 6·49-s + 5.37·67-s + 16.6·75-s − 2.88·81-s + 24.8·93-s + 6.49·97-s + 11.4·103-s + 20.5·111-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 20.7·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 7.53·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 11^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 11^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1783.579475\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1783.579475\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 - p T + 7 T^{2} - 13 T^{3} + 28 T^{4} - 13 p T^{5} + 7 p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8} )^{2} \) |
| 11 | \( 1 \) |
| good | 5 | \( ( 1 - 12 T^{2} + 113 T^{4} - 812 T^{6} + 4341 T^{8} - 812 p^{2} T^{10} + 113 p^{4} T^{12} - 12 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 7 | \( ( 1 - 3 p T^{2} + 46 p T^{4} - 3268 T^{6} + 26825 T^{8} - 3268 p^{2} T^{10} + 46 p^{5} T^{12} - 3 p^{7} T^{14} + p^{8} T^{16} )^{2} \) |
| 13 | \( ( 1 - 49 T^{2} + 1262 T^{4} - 22772 T^{6} + 326725 T^{8} - 22772 p^{2} T^{10} + 1262 p^{4} T^{12} - 49 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 17 | \( ( 1 + 56 T^{2} + 1637 T^{4} + 39868 T^{6} + 795565 T^{8} + 39868 p^{2} T^{10} + 1637 p^{4} T^{12} + 56 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 19 | \( ( 1 - 82 T^{2} + 197 p T^{4} - 114644 T^{6} + 2542285 T^{8} - 114644 p^{2} T^{10} + 197 p^{5} T^{12} - 82 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 23 | \( ( 1 - 90 T^{2} + 179 p T^{4} - 140810 T^{6} + 3753788 T^{8} - 140810 p^{2} T^{10} + 179 p^{5} T^{12} - 90 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 29 | \( ( 1 + 97 T^{2} + 5063 T^{4} + 209239 T^{6} + 6985200 T^{8} + 209239 p^{2} T^{10} + 5063 p^{4} T^{12} + 97 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 31 | \( ( 1 - 10 T + 153 T^{2} - 950 T^{3} + 7563 T^{4} - 950 p T^{5} + 153 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 37 | \( ( 1 - 9 T + 158 T^{2} - 996 T^{3} + 8953 T^{4} - 996 p T^{5} + 158 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 41 | \( ( 1 + 143 T^{2} + 9698 T^{4} + 429936 T^{6} + 17075985 T^{8} + 429936 p^{2} T^{10} + 9698 p^{4} T^{12} + 143 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 43 | \( ( 1 - 174 T^{2} + 12917 T^{4} - 562102 T^{6} + 21452940 T^{8} - 562102 p^{2} T^{10} + 12917 p^{4} T^{12} - 174 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 47 | \( ( 1 - 260 T^{2} + 33881 T^{4} - 2783020 T^{6} + 156567421 T^{8} - 2783020 p^{2} T^{10} + 33881 p^{4} T^{12} - 260 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 53 | \( ( 1 - 260 T^{2} + 35641 T^{4} - 3173420 T^{6} + 198744901 T^{8} - 3173420 p^{2} T^{10} + 35641 p^{4} T^{12} - 260 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 59 | \( ( 1 - 156 T^{2} + 15865 T^{4} - 1314404 T^{6} + 87249989 T^{8} - 1314404 p^{2} T^{10} + 15865 p^{4} T^{12} - 156 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 61 | \( ( 1 - 258 T^{2} + 35423 T^{4} - 3325396 T^{6} + 3796785 p T^{8} - 3325396 p^{2} T^{10} + 35423 p^{4} T^{12} - 258 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 67 | \( ( 1 - 11 T + 233 T^{2} - 1859 T^{3} + 23268 T^{4} - 1859 p T^{5} + 233 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 71 | \( ( 1 - 296 T^{2} + 28125 T^{4} - 283444 T^{6} - 82247131 T^{8} - 283444 p^{2} T^{10} + 28125 p^{4} T^{12} - 296 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 73 | \( ( 1 - 179 T^{2} + 25067 T^{4} - 2435617 T^{6} + 207874000 T^{8} - 2435617 p^{2} T^{10} + 25067 p^{4} T^{12} - 179 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 79 | \( ( 1 - 97 T^{2} + 15778 T^{4} - 1828404 T^{6} + 119275265 T^{8} - 1828404 p^{2} T^{10} + 15778 p^{4} T^{12} - 97 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 83 | \( ( 1 + 479 T^{2} + 109782 T^{4} + 15808912 T^{6} + 1564561325 T^{8} + 15808912 p^{2} T^{10} + 109782 p^{4} T^{12} + 479 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 89 | \( ( 1 - 154 T^{2} + 29845 T^{4} - 2842666 T^{6} + 328069564 T^{8} - 2842666 p^{2} T^{10} + 29845 p^{4} T^{12} - 154 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 97 | \( ( 1 - 16 T + 443 T^{2} - 4534 T^{3} + 66603 T^{4} - 4534 p T^{5} + 443 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
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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
−2.47747087867886515773969750530, −2.41841130467853220055865071617, −2.22970464809381599380569477194, −2.17677756500355146416549298576, −2.17262228217214637271925397471, −2.10618783000032637102477687621, −2.03876991746229687907920456807, −2.01429106440114902270677394615, −1.99224282803443532267726752086, −1.98610690901957131310897243824, −1.76071897253133705624458145859, −1.53466051292623326083480510573, −1.36823959789856701513297312468, −1.33469804420426713418615538799, −1.28850639903241312618782492817, −1.01763400526740904974901191390, −0.984520922553336864080371681809, −0.907012698353243444848480722798, −0.858162439074591353034325599323, −0.790986166353142334477447813205, −0.66388163876740529341851027068, −0.63188934827682347091977902714, −0.62708548937909336106487391885, −0.55165265208097902971213058969, −0.55073077123164262323834738509,
0.55073077123164262323834738509, 0.55165265208097902971213058969, 0.62708548937909336106487391885, 0.63188934827682347091977902714, 0.66388163876740529341851027068, 0.790986166353142334477447813205, 0.858162439074591353034325599323, 0.907012698353243444848480722798, 0.984520922553336864080371681809, 1.01763400526740904974901191390, 1.28850639903241312618782492817, 1.33469804420426713418615538799, 1.36823959789856701513297312468, 1.53466051292623326083480510573, 1.76071897253133705624458145859, 1.98610690901957131310897243824, 1.99224282803443532267726752086, 2.01429106440114902270677394615, 2.03876991746229687907920456807, 2.10618783000032637102477687621, 2.17262228217214637271925397471, 2.17677756500355146416549298576, 2.22970464809381599380569477194, 2.41841130467853220055865071617, 2.47747087867886515773969750530
Plot not available for L-functions of degree greater than 10.
