| L(s) = 1 | − 4·4-s + 8·9-s + 10·16-s + 16·23-s + 24·25-s − 32·36-s + 48·37-s − 32·53-s − 20·64-s + 48·67-s + 16·71-s + 20·81-s − 64·92-s − 96·100-s − 28·121-s + 127-s + 131-s + 137-s + 139-s + 80·144-s − 192·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 32·169-s + ⋯ |
| L(s) = 1 | − 2·4-s + 8/3·9-s + 5/2·16-s + 3.33·23-s + 24/5·25-s − 5.33·36-s + 7.89·37-s − 4.39·53-s − 5/2·64-s + 5.86·67-s + 1.89·71-s + 20/9·81-s − 6.67·92-s − 9.59·100-s − 2.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 20/3·144-s − 15.7·148-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.210162439\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.210162439\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{4} \) |
| 7 | \( 1 \) |
| 11 | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| good | 3 | \( ( 1 - 4 T^{2} + 14 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 5 | \( ( 1 - 12 T^{2} + 78 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + 16 T^{2} + 240 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + p T^{2} )^{8} \) |
| 19 | \( ( 1 + 40 T^{2} + 960 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 4 T + 32 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - 8 T^{2} - 894 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 72 T^{2} + 2640 T^{4} - 72 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 6 T + p T^{2} )^{8} \) |
| 41 | \( ( 1 + 20 T^{2} + 870 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 68 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 - 120 T^{2} + 7920 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 8 T + 50 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 - 36 T^{2} + 4974 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 208 T^{2} + 18096 T^{4} + 208 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 12 T + 138 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 2 T + p T^{2} )^{8} \) |
| 73 | \( ( 1 + 4 T^{2} + 294 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 152 T^{2} + 11616 T^{4} + 152 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 192 T^{2} + 20256 T^{4} - 192 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 288 T^{2} + 38976 T^{4} - 288 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−4.17402483941321831581903643164, −4.13915982788389670591475330769, −4.12347449994002406033147335457, −4.00505708537280522405248525662, −3.75745542300058965709636147425, −3.72210776171302024877970478964, −3.31313636448475616546623741918, −3.19372051597389921361269959937, −3.11029106768976452479271521201, −3.09748716557972960166668854051, −3.00655166299228507508366607090, −2.93572233355821301836266719111, −2.49441296138584440191662549811, −2.48763245035909541536286516443, −2.31360692479670677200779111040, −2.11601236355239761053763034663, −2.08814824533555004383526653048, −1.68587529569624291891029104163, −1.19270590449682570131363329530, −1.14550722129825163928015608082, −1.10187419935215458754528674362, −0.979977465996590940251413101929, −0.953394461067473168056892469536, −0.884864412827685459142034235266, −0.29429501799943452744540486802,
0.29429501799943452744540486802, 0.884864412827685459142034235266, 0.953394461067473168056892469536, 0.979977465996590940251413101929, 1.10187419935215458754528674362, 1.14550722129825163928015608082, 1.19270590449682570131363329530, 1.68587529569624291891029104163, 2.08814824533555004383526653048, 2.11601236355239761053763034663, 2.31360692479670677200779111040, 2.48763245035909541536286516443, 2.49441296138584440191662549811, 2.93572233355821301836266719111, 3.00655166299228507508366607090, 3.09748716557972960166668854051, 3.11029106768976452479271521201, 3.19372051597389921361269959937, 3.31313636448475616546623741918, 3.72210776171302024877970478964, 3.75745542300058965709636147425, 4.00505708537280522405248525662, 4.12347449994002406033147335457, 4.13915982788389670591475330769, 4.17402483941321831581903643164
Plot not available for L-functions of degree greater than 10.
