| L(s) = 1 | − 3·4-s − 4·7-s + 4·16-s − 18·25-s + 12·28-s + 28·31-s + 34·49-s − 9·64-s + 24·73-s − 16·79-s − 28·97-s + 54·100-s − 16·112-s − 26·121-s − 84·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s + 173-s + 72·175-s + ⋯ |
| L(s) = 1 | − 3/2·4-s − 1.51·7-s + 16-s − 3.59·25-s + 2.26·28-s + 5.02·31-s + 34/7·49-s − 9/8·64-s + 2.80·73-s − 1.80·79-s − 2.84·97-s + 27/5·100-s − 1.51·112-s − 2.36·121-s − 7.54·124-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 + 4/13·169-s + 0.0760·173-s + 5.44·175-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9678382444\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9678382444\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 3 T^{2} + 5 T^{4} + 3 p^{2} T^{6} + p^{4} T^{8} \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 + 9 T^{2} + 56 T^{4} + 9 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 - 4 T + p T^{2} )^{4}( 1 + 5 T + p T^{2} )^{4} \) |
| 11 | \( ( 1 + 13 T^{2} + 48 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 2 T^{2} - 165 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 - 18 T^{2} - 205 T^{4} - 18 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 22 T^{2} - 357 T^{4} + 22 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 11 T + p T^{2} )^{4}( 1 + 4 T + p T^{2} )^{4} \) |
| 37 | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - 54 T^{2} + 1235 T^{4} - 54 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 26 T^{2} - 1173 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 + 102 T^{2} + 6923 T^{4} + 102 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 3 T + p T^{2} )^{8} \) |
| 79 | \( ( 1 - 13 T + p T^{2} )^{4}( 1 + 17 T + p T^{2} )^{4} \) |
| 83 | \( ( 1 + 117 T^{2} + 6800 T^{4} + 117 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 66 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 + 7 T - 48 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{4} \) |
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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
−4.49584078773634706878379468877, −4.38256923011874471299996721401, −4.22942312961620754110201103796, −4.20892004791138526396399445679, −4.14023886830941533861024067631, −3.91009289835576112350472433324, −3.82647498992848099545548303974, −3.71181534504462240383229583980, −3.48765857033127964583721813185, −3.46948379640477882700849133934, −3.05039987885075078839393874774, −2.91491504242085576772986753048, −2.91238376205532063690637598470, −2.85683482669158158497628989621, −2.63163654082786437401444656066, −2.25376630075413671866525897956, −2.24525093534973007623253565392, −2.01113442614550936147604748116, −2.00538733315545875183692457873, −1.55695901593945557337494238349, −1.23763618543520277559053537568, −1.05010596554921497192628901346, −0.794904761691422534358901544123, −0.57752701139125380400912269644, −0.22913639274563606743476023325,
0.22913639274563606743476023325, 0.57752701139125380400912269644, 0.794904761691422534358901544123, 1.05010596554921497192628901346, 1.23763618543520277559053537568, 1.55695901593945557337494238349, 2.00538733315545875183692457873, 2.01113442614550936147604748116, 2.24525093534973007623253565392, 2.25376630075413671866525897956, 2.63163654082786437401444656066, 2.85683482669158158497628989621, 2.91238376205532063690637598470, 2.91491504242085576772986753048, 3.05039987885075078839393874774, 3.46948379640477882700849133934, 3.48765857033127964583721813185, 3.71181534504462240383229583980, 3.82647498992848099545548303974, 3.91009289835576112350472433324, 4.14023886830941533861024067631, 4.20892004791138526396399445679, 4.22942312961620754110201103796, 4.38256923011874471299996721401, 4.49584078773634706878379468877
Plot not available for L-functions of degree greater than 10.
