| L(s) = 1 | + 4·7-s + 4·13-s − 56·19-s − 64·25-s − 32·31-s + 112·37-s + 52·43-s + 148·49-s + 64·61-s − 20·67-s − 512·73-s − 308·79-s + 16·91-s + 160·97-s + 448·103-s + 376·109-s − 232·121-s + 127-s + 131-s − 224·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 4/7·7-s + 4/13·13-s − 2.94·19-s − 2.55·25-s − 1.03·31-s + 3.02·37-s + 1.20·43-s + 3.02·49-s + 1.04·61-s − 0.298·67-s − 7.01·73-s − 3.89·79-s + 0.175·91-s + 1.64·97-s + 4.34·103-s + 3.44·109-s − 1.91·121-s + 0.00787·127-s + 0.00763·131-s − 1.68·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.182120785\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.182120785\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 64 T^{2} + 413 p T^{4} + 49984 T^{6} + 1174336 T^{8} + 49984 p^{4} T^{10} + 413 p^{9} T^{12} + 64 p^{12} T^{14} + p^{16} T^{16} \) |
| 7 | \( ( 1 - 2 T - 68 T^{2} + 52 T^{3} + 2587 T^{4} + 52 p^{2} T^{5} - 68 p^{4} T^{6} - 2 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 11 | \( 1 + 232 T^{2} + 26638 T^{4} - 486272 T^{6} - 263317469 T^{8} - 486272 p^{4} T^{10} + 26638 p^{8} T^{12} + 232 p^{12} T^{14} + p^{16} T^{16} \) |
| 13 | \( ( 1 - 2 T - 227 T^{2} + 214 T^{3} + 24124 T^{4} + 214 p^{2} T^{5} - 227 p^{4} T^{6} - 2 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 + 392 T^{2} + 164391 T^{4} + 392 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 + 14 T + 96 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 23 | \( 1 + 352 T^{2} - 404546 T^{4} - 10993664 T^{6} + 166274997955 T^{8} - 10993664 p^{4} T^{10} - 404546 p^{8} T^{12} + 352 p^{12} T^{14} + p^{16} T^{16} \) |
| 29 | \( 1 - 344 T^{2} + 114937 T^{4} + 485440072 T^{6} - 583931664176 T^{8} + 485440072 p^{4} T^{10} + 114937 p^{8} T^{12} - 344 p^{12} T^{14} + p^{16} T^{16} \) |
| 31 | \( ( 1 + 16 T - 758 T^{2} - 14528 T^{3} - 141341 T^{4} - 14528 p^{2} T^{5} - 758 p^{4} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 - 28 T + 2907 T^{2} - 28 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 41 | \( 1 + 1720 T^{2} + 2788078 T^{4} - 9427664000 T^{6} - 16315646987837 T^{8} - 9427664000 p^{4} T^{10} + 2788078 p^{8} T^{12} + 1720 p^{12} T^{14} + p^{16} T^{16} \) |
| 43 | \( ( 1 - 26 T - 1004 T^{2} + 52468 T^{3} - 1846325 T^{4} + 52468 p^{2} T^{5} - 1004 p^{4} T^{6} - 26 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 47 | \( 1 + 4948 T^{2} + 11437018 T^{4} + 16260731152 T^{6} + 26910851011411 T^{8} + 16260731152 p^{4} T^{10} + 11437018 p^{8} T^{12} + 4948 p^{12} T^{14} + p^{16} T^{16} \) |
| 53 | \( ( 1 - 7960 T^{2} + 30066162 T^{4} - 7960 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 59 | \( 1 + 6868 T^{2} + 22903546 T^{4} + 213979408 T^{6} - 129566541263501 T^{8} + 213979408 p^{4} T^{10} + 22903546 p^{8} T^{12} + 6868 p^{12} T^{14} + p^{16} T^{16} \) |
| 61 | \( ( 1 - 32 T - 4487 T^{2} + 61792 T^{3} + 12714976 T^{4} + 61792 p^{2} T^{5} - 4487 p^{4} T^{6} - 32 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 + 10 T - 8660 T^{2} - 2180 T^{3} + 56137579 T^{4} - 2180 p^{2} T^{5} - 8660 p^{4} T^{6} + 10 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 - 7168 T^{2} + 21615810 T^{4} - 7168 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 + 128 T + 12567 T^{2} + 128 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 79 | \( ( 1 + 154 T + 6628 T^{2} + 709324 T^{3} + 105038107 T^{4} + 709324 p^{2} T^{5} + 6628 p^{4} T^{6} + 154 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 83 | \( 1 + 340 T^{2} + 86568586 T^{4} - 61665673520 T^{6} + 5216340374269555 T^{8} - 61665673520 p^{4} T^{10} + 86568586 p^{8} T^{12} + 340 p^{12} T^{14} + p^{16} T^{16} \) |
| 89 | \( ( 1 - 19120 T^{2} + 179727999 T^{4} - 19120 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 - 80 T - 950 T^{2} + 917440 T^{3} - 94724381 T^{4} + 917440 p^{2} T^{5} - 950 p^{4} T^{6} - 80 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
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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
−4.93743436570500794673182601322, −4.67626562594540861416013220431, −4.51441574109500847156369189227, −4.39032434749212114539504659797, −4.31753499944601869780977102086, −4.17233561316921489330267223862, −4.06793845613935937100827739226, −3.95408064236994023297875798467, −3.83546310042165985967292192158, −3.57772036723962910700712030798, −3.53673459395072288819965077763, −3.03120563758690001492982968720, −2.89582744796415355141147947983, −2.84596745672364431325722131405, −2.49029444710459692127393260877, −2.48800647093926439036644607934, −2.39001823642534088626382045758, −1.95099200821871735811686655464, −1.81446650370707381432019965373, −1.74831102416278600532033768098, −1.43844649256055550534873133676, −1.03009884883773354930612710232, −0.977685382070878909114443237912, −0.36084998951477977683199397301, −0.17724233507019846450033173011,
0.17724233507019846450033173011, 0.36084998951477977683199397301, 0.977685382070878909114443237912, 1.03009884883773354930612710232, 1.43844649256055550534873133676, 1.74831102416278600532033768098, 1.81446650370707381432019965373, 1.95099200821871735811686655464, 2.39001823642534088626382045758, 2.48800647093926439036644607934, 2.49029444710459692127393260877, 2.84596745672364431325722131405, 2.89582744796415355141147947983, 3.03120563758690001492982968720, 3.53673459395072288819965077763, 3.57772036723962910700712030798, 3.83546310042165985967292192158, 3.95408064236994023297875798467, 4.06793845613935937100827739226, 4.17233561316921489330267223862, 4.31753499944601869780977102086, 4.39032434749212114539504659797, 4.51441574109500847156369189227, 4.67626562594540861416013220431, 4.93743436570500794673182601322
Plot not available for L-functions of degree greater than 10.
