| L(s) = 1 | − 4·7-s − 2·16-s − 16·31-s − 24·37-s + 24·43-s + 8·49-s + 48·61-s + 8·67-s − 68·73-s + 60·97-s + 64·103-s + 8·112-s + 32·121-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 + ⋯ |
| L(s) = 1 | − 1.51·7-s − 1/2·16-s − 2.87·31-s − 3.94·37-s + 3.65·43-s + 8/7·49-s + 6.14·61-s + 0.977·67-s − 7.95·73-s + 6.09·97-s + 6.30·103-s + 0.755·112-s + 2.90·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.814955468\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.814955468\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{4} )^{2} \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 + 2 T + 2 T^{2} + 12 T^{3} + 71 T^{4} + 12 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 16 T^{2} + 21 p T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 334 T^{4} + p^{4} T^{8} )^{2} \) |
| 17 | \( 1 - 124 T^{4} - 77946 T^{8} - 124 p^{4} T^{12} + p^{8} T^{16} \) |
| 19 | \( ( 1 - 20 T^{2} + 54 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 706 T^{4} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 64 T^{2} + 2514 T^{4} + 64 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 4 T - 9 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + 12 T + 72 T^{2} + 12 T^{3} - 1294 T^{4} + 12 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 52 T^{2} + 2838 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 12 T + 72 T^{2} - 660 T^{3} + 5906 T^{4} - 660 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 1054 T^{4} + p^{4} T^{8} )^{2} \) |
| 53 | \( 1 + 878 T^{4} + 6253683 T^{8} + 878 p^{4} T^{12} + p^{8} T^{16} \) |
| 59 | \( ( 1 + 112 T^{2} + 8370 T^{4} + 112 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 12 T + 146 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 4 T + 8 T^{2} - 60 T^{3} - 2254 T^{4} - 60 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 28 T^{2} - 2010 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 17 T + p T^{2} )^{4}( 1 + 143 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | \( ( 1 - 146 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( 1 + 6286 T^{4} + 19632339 T^{8} + 6286 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( ( 1 + 244 T^{2} + 30294 T^{4} + 244 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 30 T + 450 T^{2} - 4080 T^{3} + 35471 T^{4} - 4080 p T^{5} + 450 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−3.88569984962702533978880007744, −3.86526893820192050926517767123, −3.76264399209538606951995578161, −3.72146338419279997867683997781, −3.69664873482844562861093778945, −3.36785063954533609870521789446, −3.33327990843366005362523510335, −3.29851389502508644577260425038, −3.15267955399192766231155273181, −2.99032685533505186678124107982, −2.60850162206498348692226594100, −2.55321983511391091952032180108, −2.45918251238146373361815162972, −2.39859121767611385391105765859, −2.25944493970122890474708629635, −2.08195279556612348396631170275, −1.70440623496560762538131849006, −1.68404400516956108229716674765, −1.61859404661630340489084635154, −1.61114214320322358257257355579, −1.03762372211956648975650616143, −0.73340809871335329737642347724, −0.64834530101153054332666186568, −0.52989045453619554196041620153, −0.23148371473036718811291667978,
0.23148371473036718811291667978, 0.52989045453619554196041620153, 0.64834530101153054332666186568, 0.73340809871335329737642347724, 1.03762372211956648975650616143, 1.61114214320322358257257355579, 1.61859404661630340489084635154, 1.68404400516956108229716674765, 1.70440623496560762538131849006, 2.08195279556612348396631170275, 2.25944493970122890474708629635, 2.39859121767611385391105765859, 2.45918251238146373361815162972, 2.55321983511391091952032180108, 2.60850162206498348692226594100, 2.99032685533505186678124107982, 3.15267955399192766231155273181, 3.29851389502508644577260425038, 3.33327990843366005362523510335, 3.36785063954533609870521789446, 3.69664873482844562861093778945, 3.72146338419279997867683997781, 3.76264399209538606951995578161, 3.86526893820192050926517767123, 3.88569984962702533978880007744
Plot not available for L-functions of degree greater than 10.
