| L(s) = 1 | + 4·3-s − 7·4-s + 4·7-s + 9·9-s − 28·12-s − 32·13-s + 16·16-s − 112·19-s + 16·21-s − 46·25-s − 28·28-s − 18·31-s − 63·36-s − 120·37-s − 128·39-s + 264·43-s + 64·48-s − 18·49-s + 224·52-s − 448·57-s − 336·61-s + 36·63-s − 24·67-s + 182·73-s − 184·75-s + 784·76-s + 460·79-s + ⋯ |
| L(s) = 1 | + 4/3·3-s − 7/4·4-s + 4/7·7-s + 9-s − 7/3·12-s − 2.46·13-s + 16-s − 5.89·19-s + 0.761·21-s − 1.83·25-s − 28-s − 0.580·31-s − 7/4·36-s − 3.24·37-s − 3.28·39-s + 6.13·43-s + 4/3·48-s − 0.367·49-s + 4.30·52-s − 7.85·57-s − 5.50·61-s + 4/7·63-s − 0.358·67-s + 2.49·73-s − 2.45·75-s + 10.3·76-s + 5.82·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.074972673\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.074972673\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 4 T + 7 T^{2} + 8 T^{3} - 95 T^{4} + 8 p^{2} T^{5} + 7 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 7 T^{2} + 33 T^{4} + 119 T^{6} + 305 T^{8} + 119 p^{4} T^{10} + 33 p^{8} T^{12} + 7 p^{12} T^{14} + p^{16} T^{16} \) |
| 5 | \( 1 + 46 T^{2} + 171 T^{4} - 24844 T^{6} - 756019 T^{8} - 24844 p^{4} T^{10} + 171 p^{8} T^{12} + 46 p^{12} T^{14} + p^{16} T^{16} \) |
| 7 | \( ( 1 - 2 T + 15 T^{2} - 292 T^{3} + 2909 T^{4} - 292 p^{2} T^{5} + 15 p^{4} T^{6} - 2 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 + 16 T - 33 T^{2} - 3022 T^{3} - 37075 T^{4} - 3022 p^{2} T^{5} - 33 p^{4} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 17 | \( 1 + 542 T^{2} + 210243 T^{4} + 68683324 T^{6} + 19666656005 T^{8} + 68683324 p^{4} T^{10} + 210243 p^{8} T^{12} + 542 p^{12} T^{14} + p^{16} T^{16} \) |
| 19 | \( ( 1 + 56 T + 975 T^{2} + 1714 T^{3} - 131251 T^{4} + 1714 p^{2} T^{5} + 975 p^{4} T^{6} + 56 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 23 | \( ( 1 - 1108 T^{2} + 827878 T^{4} - 1108 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 29 | \( 1 + 67 p T^{2} + 1137963 T^{4} + 548468221 T^{6} + 624074488400 T^{8} + 548468221 p^{4} T^{10} + 1137963 p^{8} T^{12} + 67 p^{13} T^{14} + p^{16} T^{16} \) |
| 31 | \( ( 1 + 9 T - 165 T^{2} + 26771 T^{3} + 1096464 T^{4} + 26771 p^{2} T^{5} - 165 p^{4} T^{6} + 9 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 + 60 T + 471 T^{2} + 15310 T^{3} + 2183181 T^{4} + 15310 p^{2} T^{5} + 471 p^{4} T^{6} + 60 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 41 | \( 1 + 4486 T^{2} + 5535915 T^{4} - 12051684736 T^{6} - 42824146054651 T^{8} - 12051684736 p^{4} T^{10} + 5535915 p^{8} T^{12} + 4486 p^{12} T^{14} + p^{16} T^{16} \) |
| 43 | \( ( 1 - 66 T + 4742 T^{2} - 66 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 47 | \( 1 + 3922 T^{2} + 13265163 T^{4} + 41243684144 T^{6} + 99026272083365 T^{8} + 41243684144 p^{4} T^{10} + 13265163 p^{8} T^{12} + 3922 p^{12} T^{14} + p^{16} T^{16} \) |
| 53 | \( 1 + 5719 T^{2} + 20370555 T^{4} + 53901639581 T^{6} + 152432594812784 T^{8} + 53901639581 p^{4} T^{10} + 20370555 p^{8} T^{12} + 5719 p^{12} T^{14} + p^{16} T^{16} \) |
| 59 | \( 1 + 6862 T^{2} + 26290683 T^{4} + 1000557196 p T^{6} + 105471828660605 T^{8} + 1000557196 p^{5} T^{10} + 26290683 p^{8} T^{12} + 6862 p^{12} T^{14} + p^{16} T^{16} \) |
| 61 | \( ( 1 + 168 T + 8303 T^{2} + 15126 T^{3} - 11368595 T^{4} + 15126 p^{2} T^{5} + 8303 p^{4} T^{6} + 168 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 + 6 T + 3542 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 71 | \( 1 + 178 T^{2} - 293397 T^{4} - 100251984784 T^{6} + 448183018648805 T^{8} - 100251984784 p^{4} T^{10} - 293397 p^{8} T^{12} + 178 p^{12} T^{14} + p^{16} T^{16} \) |
| 73 | \( ( 1 - 91 T - 2193 T^{2} + 423367 T^{3} - 13036600 T^{4} + 423367 p^{2} T^{5} - 2193 p^{4} T^{6} - 91 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 79 | \( ( 1 - 230 T + 19119 T^{2} - 959620 T^{3} + 61893701 T^{4} - 959620 p^{2} T^{5} + 19119 p^{4} T^{6} - 230 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 83 | \( 1 + 1774 T^{2} - 45091365 T^{4} - 141925594684 T^{6} + 1890623755738109 T^{8} - 141925594684 p^{4} T^{10} - 45091365 p^{8} T^{12} + 1774 p^{12} T^{14} + p^{16} T^{16} \) |
| 89 | \( ( 1 - 22052 T^{2} + 224754438 T^{4} - 22052 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 - 258 T + 27455 T^{2} - 2679828 T^{3} + 299189869 T^{4} - 2679828 p^{2} T^{5} + 27455 p^{4} T^{6} - 258 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
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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
−4.59232265584124644599423541485, −4.57913529134520048819335269568, −4.57814369378862689872831357758, −4.41175519137631510980357046468, −4.40677826639980606983814481967, −4.19831052568556532710124200183, −3.95581446507364059667802904457, −3.68009565774097883106351302957, −3.66873733011274839215034294965, −3.55154876163977608092995270799, −3.40656287086764705207513504617, −3.09128236439213175509102783870, −3.08645633535326335199908581920, −2.45208600824561780185174962645, −2.44038790835799665677441028896, −2.39766242532343524602520411781, −2.14145280519070058786798700578, −2.06295651462768903784292377008, −1.91421865351865709675656275078, −1.76597990416535887635032753676, −1.67604541214683839060389185130, −0.830217798821830990583132007294, −0.57537853406444596400849087923, −0.35566638566095665706264477729, −0.32008759594524802823339774594,
0.32008759594524802823339774594, 0.35566638566095665706264477729, 0.57537853406444596400849087923, 0.830217798821830990583132007294, 1.67604541214683839060389185130, 1.76597990416535887635032753676, 1.91421865351865709675656275078, 2.06295651462768903784292377008, 2.14145280519070058786798700578, 2.39766242532343524602520411781, 2.44038790835799665677441028896, 2.45208600824561780185174962645, 3.08645633535326335199908581920, 3.09128236439213175509102783870, 3.40656287086764705207513504617, 3.55154876163977608092995270799, 3.66873733011274839215034294965, 3.68009565774097883106351302957, 3.95581446507364059667802904457, 4.19831052568556532710124200183, 4.40677826639980606983814481967, 4.41175519137631510980357046468, 4.57814369378862689872831357758, 4.57913529134520048819335269568, 4.59232265584124644599423541485
Plot not available for L-functions of degree greater than 10.
