| L(s) = 1 | + 16·7-s + 20·13-s − 2·16-s − 8·25-s − 8·31-s + 24·37-s + 128·49-s − 16·61-s + 36·67-s − 48·73-s − 18·81-s + 320·91-s − 60·97-s − 16·103-s − 32·112-s + 32·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 200·169-s + 173-s + ⋯ |
| L(s) = 1 | + 6.04·7-s + 5.54·13-s − 1/2·16-s − 8/5·25-s − 1.43·31-s + 3.94·37-s + 18.2·49-s − 2.04·61-s + 4.39·67-s − 5.61·73-s − 2·81-s + 33.5·91-s − 6.09·97-s − 1.57·103-s − 3.02·112-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 + 15.3·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(37.93414456\) |
| \(L(\frac12)\) |
\(\approx\) |
\(37.93414456\) |
| \(L(\frac{3}{2})\) |
|
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 | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 5 | \( 1 + 8 T^{2} + 39 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | \( ( 1 + T )^{8} \) |
| good | 7 | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 - 16 T^{2} + 21 p T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | \( ( 1 + 511 T^{4} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 29 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 + 98 T^{4} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 4 T^{2} + 486 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 12 T + 72 T^{2} - 372 T^{3} + 1886 T^{4} - 372 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 + 146 T^{4} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 + 6398 T^{4} + 18539715 T^{8} + 6398 p^{4} T^{12} + p^{8} T^{16} \) |
| 53 | \( ( 1 - 2254 T^{4} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 124 T^{2} + 9606 T^{4} + 124 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 18 T + 162 T^{2} - 1260 T^{3} + 9791 T^{4} - 1260 p T^{5} + 162 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 40 T^{2} - 2193 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 24 T + 288 T^{2} + 2904 T^{3} + 26978 T^{4} + 2904 p T^{5} + 288 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 302 T^{2} + 35235 T^{4} - 302 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 6418 T^{4} + 75105651 T^{8} - 6418 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( ( 1 + 80 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 + 30 T + 450 T^{2} + 6240 T^{3} + 74207 T^{4} + 6240 p T^{5} + 450 p^{2} T^{6} + 30 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−4.26859777001398957237995381179, −4.19002162575539566522971746659, −4.14975675347117337125165469115, −4.14084141125305610164538372945, −3.93316079955597113027209424657, −3.66319610805428526333508266189, −3.64455370033652090238233428316, −3.51696839423593325724463480947, −3.27039748438159743064705793936, −3.25654876085376603150565709073, −2.80868022738189471318854129617, −2.69224575136385437313890772517, −2.60110090098288527662325285209, −2.52379759403154234472076293527, −2.26784404680484281523051113999, −2.07947240219472408664006697898, −1.66048304738061717328280872134, −1.65484113387145307355211507907, −1.64234347350837857110472166922, −1.53794715201500429327008880443, −1.33166042201689597615221631131, −1.21354039676173014338116008607, −1.19735092118363966755840722823, −0.71928242065889306004602204468, −0.49687332484574566474842773611,
0.49687332484574566474842773611, 0.71928242065889306004602204468, 1.19735092118363966755840722823, 1.21354039676173014338116008607, 1.33166042201689597615221631131, 1.53794715201500429327008880443, 1.64234347350837857110472166922, 1.65484113387145307355211507907, 1.66048304738061717328280872134, 2.07947240219472408664006697898, 2.26784404680484281523051113999, 2.52379759403154234472076293527, 2.60110090098288527662325285209, 2.69224575136385437313890772517, 2.80868022738189471318854129617, 3.25654876085376603150565709073, 3.27039748438159743064705793936, 3.51696839423593325724463480947, 3.64455370033652090238233428316, 3.66319610805428526333508266189, 3.93316079955597113027209424657, 4.14084141125305610164538372945, 4.14975675347117337125165469115, 4.19002162575539566522971746659, 4.26859777001398957237995381179
Plot not available for L-functions of degree greater than 10.
