| L(s) = 1 | − 8·25-s − 36·49-s + 32·59-s − 8·89-s + 56·101-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 48·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
| L(s) = 1 | − 8/5·25-s − 5.14·49-s + 4.16·59-s − 0.847·89-s + 5.57·101-s + 4/11·121-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 + 3.69·169-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 + 0.0663·227-s + 0.0660·229-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.535258094\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.535258094\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 8 T^{2} + 38 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | \( 1 + 44 T^{2} + 950 T^{4} + 44 p^{2} T^{6} + p^{4} T^{8} \) |
| good | 7 | \( ( 1 + 18 T^{2} + 172 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 2 T^{2} + 180 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 24 T^{2} + 370 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 + 38 T^{2} + 1412 T^{4} + 38 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - p T^{2} )^{8} \) |
| 31 | \( ( 1 - 54 T^{2} + 1804 T^{4} - 54 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 108 T^{2} + 5542 T^{4} + 108 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 136 T^{2} + 7958 T^{4} - 136 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 88 T^{2} + 4626 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 8 T^{2} + 1634 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 100 T^{2} + 8006 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 8 T + 106 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 48 T^{2} - 74 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 72 T^{2} + 7474 T^{4} - 72 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 130 T^{2} + 11780 T^{4} - 130 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 48 T^{2} - 3578 T^{4} + 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 246 T^{2} + 26764 T^{4} - 246 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 - 80 T^{2} + 6306 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 + 180 T^{2} + 26470 T^{4} + 180 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−3.55000921922029191720387706156, −3.54462033935359468088559935603, −3.44320874160619442661050321749, −3.25528668527822376232048329248, −3.15768289707137016259084411478, −3.03969881149047152798517230902, −2.94049950909660866895379429726, −2.87146946408927394106384304370, −2.61165592157803413363134331651, −2.54255501580027400771890228389, −2.25969277639122780563558507575, −2.23967228992006668422758776983, −2.12180693163440482177221840916, −2.09393100268033922473790198081, −1.79648086728268565056700691557, −1.79440818919450939512448778593, −1.59871754882578056993946413971, −1.54893811682610370259077689902, −1.32674320024536566170721942336, −1.04821480040811486021504810395, −0.984723615740513249053881910345, −0.68725172001084567428913164553, −0.46637233613722196681150968776, −0.46322037813197485693600109639, −0.20773165934077354986605102968,
0.20773165934077354986605102968, 0.46322037813197485693600109639, 0.46637233613722196681150968776, 0.68725172001084567428913164553, 0.984723615740513249053881910345, 1.04821480040811486021504810395, 1.32674320024536566170721942336, 1.54893811682610370259077689902, 1.59871754882578056993946413971, 1.79440818919450939512448778593, 1.79648086728268565056700691557, 2.09393100268033922473790198081, 2.12180693163440482177221840916, 2.23967228992006668422758776983, 2.25969277639122780563558507575, 2.54255501580027400771890228389, 2.61165592157803413363134331651, 2.87146946408927394106384304370, 2.94049950909660866895379429726, 3.03969881149047152798517230902, 3.15768289707137016259084411478, 3.25528668527822376232048329248, 3.44320874160619442661050321749, 3.54462033935359468088559935603, 3.55000921922029191720387706156
Plot not available for L-functions of degree greater than 10.
