| L(s) = 1 | − 32·25-s + 80·49-s − 32·73-s + 32·97-s + 32·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | − 6.39·25-s + 80/7·49-s − 3.74·73-s + 3.24·97-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 − 0.615·169-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 + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{96} \cdot 3^{64}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{96} \cdot 3^{64}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6852222819\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6852222819\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 + 8 T^{2} + 63 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 7 | \( ( 1 - 20 T^{2} + 186 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 11 | \( ( 1 - 8 T^{2} + 6 p T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 13 | \( ( 1 + 2 T^{2} - 93 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 17 | \( ( 1 - 32 T^{2} + 591 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 19 | \( ( 1 + 4 T^{2} + 618 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 23 | \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{8} \) |
| 29 | \( ( 1 + 80 T^{2} + 3255 T^{4} + 80 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 31 | \( ( 1 - 92 T^{2} + 3846 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 37 | \( ( 1 - 58 T^{2} + 1851 T^{4} - 58 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 41 | \( ( 1 - 28 T^{2} + p^{2} T^{4} )^{8} \) |
| 43 | \( ( 1 + 100 T^{2} + 6090 T^{4} + 100 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 47 | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{8} \) |
| 53 | \( ( 1 + 152 T^{2} + 10626 T^{4} + 152 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 59 | \( ( 1 - 92 T^{2} + 8646 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 61 | \( ( 1 - 82 T^{2} + 5235 T^{4} - 82 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 67 | \( ( 1 + 52 T^{2} + 8682 T^{4} + 52 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 71 | \( ( 1 + 88 T^{2} + p^{2} T^{4} )^{8} \) |
| 73 | \( ( 1 + 4 T + 75 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{8} \) |
| 79 | \( ( 1 - 260 T^{2} + 29274 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 83 | \( ( 1 - 92 T^{2} + 2022 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 89 | \( ( 1 - 104 T^{2} + 16359 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 97 | \( ( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.03173180283446277055344575118, −2.01314240512950205302982352756, −1.77329413554424098326266993187, −1.66652774880485318401691699118, −1.66243639897497677681990402988, −1.65194030431893030002451447681, −1.64597278151155120206353667843, −1.50651068778611121944721560351, −1.49972967301410695354596505127, −1.33056807722844270919176904521, −1.23391050486015418247922120817, −1.19418449046014441402830081986, −1.15696619227370493181200538919, −1.04486121360685577032221400055, −0.922815753317448144271367235006, −0.802919651108320211102809229465, −0.73344947024762219738591633053, −0.67972549372067330994978982557, −0.65958065435556590809555704431, −0.48199492172959556728596972072, −0.45219541732824348520184278138, −0.37852166100866393320027822042, −0.36462044232126049118004794876, −0.32347382003653234524290818341, −0.01404605343370135246433892380,
0.01404605343370135246433892380, 0.32347382003653234524290818341, 0.36462044232126049118004794876, 0.37852166100866393320027822042, 0.45219541732824348520184278138, 0.48199492172959556728596972072, 0.65958065435556590809555704431, 0.67972549372067330994978982557, 0.73344947024762219738591633053, 0.802919651108320211102809229465, 0.922815753317448144271367235006, 1.04486121360685577032221400055, 1.15696619227370493181200538919, 1.19418449046014441402830081986, 1.23391050486015418247922120817, 1.33056807722844270919176904521, 1.49972967301410695354596505127, 1.50651068778611121944721560351, 1.64597278151155120206353667843, 1.65194030431893030002451447681, 1.66243639897497677681990402988, 1.66652774880485318401691699118, 1.77329413554424098326266993187, 2.01314240512950205302982352756, 2.03173180283446277055344575118
Plot not available for L-functions of degree greater than 10.
