| L(s) = 1 | + 16·9-s − 12·11-s + 4·19-s − 24·29-s − 48·59-s + 16·61-s + 48·71-s − 24·79-s + 130·81-s − 96·89-s − 192·99-s − 24·101-s − 8·109-s + 72·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 8·169-s + 64·171-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 16/3·9-s − 3.61·11-s + 0.917·19-s − 4.45·29-s − 6.24·59-s + 2.04·61-s + 5.69·71-s − 2.70·79-s + 14.4·81-s − 10.1·89-s − 19.2·99-s − 2.38·101-s − 0.766·109-s + 6.54·121-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 − 0.615·169-s + 4.89·171-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.02478798437\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02478798437\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( ( 1 - 8 T^{2} + 31 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 7 | \( 1 + 4 p T^{4} + 3270 T^{8} + 4 p^{5} T^{12} + p^{8} T^{16} \) |
| 11 | \( ( 1 + 6 T + 18 T^{2} + 12 T^{3} - 73 T^{4} + 12 p T^{5} + 18 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 + 511 T^{4} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 2 T + 2 T^{2} - 12 T^{3} - 97 T^{4} - 12 p T^{5} + 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( 1 - 1252 T^{4} + 811590 T^{8} - 1252 p^{4} T^{12} + p^{8} T^{16} \) |
| 29 | \( ( 1 + 12 T + 72 T^{2} + 492 T^{3} + 3218 T^{4} + 492 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 68 T^{2} + 2970 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 8 T^{2} + 1026 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 122 T^{2} + 6651 T^{4} - 122 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 88 T^{2} + 3906 T^{4} + 88 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 3682 T^{4} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 + 24 T + 288 T^{2} + 3000 T^{3} + 26894 T^{4} + 3000 p T^{5} + 288 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 8 T + 32 T^{2} - 120 T^{3} - 1666 T^{4} - 120 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + 232 T^{2} + 22191 T^{4} + 232 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 12 T + 130 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 73 | \( 1 - 7922 T^{4} + 28800435 T^{8} - 7922 p^{4} T^{12} + p^{8} T^{16} \) |
| 79 | \( ( 1 + 6 T + 140 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 8 T^{2} - 5889 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 24 T + 319 T^{2} + 24 p T^{3} + p^{2} T^{4} )^{4} \) |
| 97 | \( 1 - 1532 T^{4} + 22125318 T^{8} - 1532 p^{4} T^{12} + p^{8} T^{16} \) |
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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
−3.98013354633739556855707058744, −3.96196855816044345503029843564, −3.67765584920306994526957284286, −3.61380210835015800415456307327, −3.58267999649906873905744379826, −3.57532341250881269850627576044, −3.02178755202013784108782686492, −2.97414894909233668343380497104, −2.94682584460696345730738001773, −2.87987500721494508316613651643, −2.81640232231879661638541983746, −2.56998853185724738185261593338, −2.29877182135251106157050524498, −2.11549448888050874483016525523, −2.01585870638201445859092585670, −2.00088810301461538832190178576, −1.69935127921607798180273928483, −1.65801333279386469218838824719, −1.49202865058545096507475739307, −1.40124506754617687162271518072, −1.13812648872713016424791088741, −0.979269419352819304230081351144, −0.909296249117543762294682115797, −0.14731205732636325815223493706, −0.03596590167471780336196843149,
0.03596590167471780336196843149, 0.14731205732636325815223493706, 0.909296249117543762294682115797, 0.979269419352819304230081351144, 1.13812648872713016424791088741, 1.40124506754617687162271518072, 1.49202865058545096507475739307, 1.65801333279386469218838824719, 1.69935127921607798180273928483, 2.00088810301461538832190178576, 2.01585870638201445859092585670, 2.11549448888050874483016525523, 2.29877182135251106157050524498, 2.56998853185724738185261593338, 2.81640232231879661638541983746, 2.87987500721494508316613651643, 2.94682584460696345730738001773, 2.97414894909233668343380497104, 3.02178755202013784108782686492, 3.57532341250881269850627576044, 3.58267999649906873905744379826, 3.61380210835015800415456307327, 3.67765584920306994526957284286, 3.96196855816044345503029843564, 3.98013354633739556855707058744
Plot not available for L-functions of degree greater than 10.
