| L(s) = 1 | + 10·9-s + 26·25-s + 20·43-s + 12·53-s + 20·67-s + 49·81-s + 40·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 32·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 260·225-s + ⋯ |
| L(s) = 1 | + 10/3·9-s + 26/5·25-s + 3.04·43-s + 1.64·53-s + 2.44·67-s + 49/9·81-s + 3.63·121-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 − 2.46·169-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 + 52/3·225-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{16} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{16} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(24.36092484\) |
| \(L(\frac12)\) |
\(\approx\) |
\(24.36092484\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| good | 3 | \( ( 1 - 5 T^{2} + 13 T^{4} - 5 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 5 | \( ( 1 - 13 T^{2} + 81 T^{4} - 13 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 + 16 T^{2} + 222 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 40 T^{2} + 942 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 56 T^{2} + 1662 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 97 T^{2} + 4173 T^{4} - 97 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 64 T^{2} + 2142 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 157 T^{2} + 9513 T^{4} - 157 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 5 T + 81 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 + 82 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 - 3 T + 97 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 37 T^{2} + 6873 T^{4} - 37 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 5 T + 39 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 200 T^{2} + 18462 T^{4} - 200 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 157 T^{2} + 14289 T^{4} - 157 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 16 T + p T^{2} )^{4}( 1 + 16 T + p T^{2} )^{4} \) |
| 83 | \( ( 1 + 56 T^{2} + 12942 T^{4} + 56 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 325 T^{2} + 44313 T^{4} - 325 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−3.67043224649599321921372115004, −3.37536053348820213565971237135, −3.29822668015596840147180587136, −3.28409355225779754273054058950, −3.20769780129973526521572528344, −3.04248917314629386665027820156, −2.71849793775440241749226796441, −2.55088805469103063641341980893, −2.51090948392395278511446753422, −2.50736718298712108298920679475, −2.50485754775703819181263966520, −2.36802483371766203422864958819, −2.21840393606828419464700221535, −2.03119810727962149375962063387, −1.66745383720972612140427375616, −1.52875442121550740571644940052, −1.45511767795379466034053490702, −1.45399751469632690037250208671, −1.32086829240605591844262954489, −1.03705599490888913484608140844, −0.976998964330555235947202580574, −0.858384588681719640005845997532, −0.63354242243175169220054465514, −0.59658060743173226785314336543, −0.21991285521800350357138704208,
0.21991285521800350357138704208, 0.59658060743173226785314336543, 0.63354242243175169220054465514, 0.858384588681719640005845997532, 0.976998964330555235947202580574, 1.03705599490888913484608140844, 1.32086829240605591844262954489, 1.45399751469632690037250208671, 1.45511767795379466034053490702, 1.52875442121550740571644940052, 1.66745383720972612140427375616, 2.03119810727962149375962063387, 2.21840393606828419464700221535, 2.36802483371766203422864958819, 2.50485754775703819181263966520, 2.50736718298712108298920679475, 2.51090948392395278511446753422, 2.55088805469103063641341980893, 2.71849793775440241749226796441, 3.04248917314629386665027820156, 3.20769780129973526521572528344, 3.28409355225779754273054058950, 3.29822668015596840147180587136, 3.37536053348820213565971237135, 3.67043224649599321921372115004
Plot not available for L-functions of degree greater than 10.
