| L(s) = 1 | + 20·25-s + 72·41-s − 20·49-s − 64·73-s + 8·97-s − 72·113-s − 76·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 68·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | + 4·25-s + 11.2·41-s − 2.85·49-s − 7.49·73-s + 0.812·97-s − 6.77·113-s − 6.90·121-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 − 5.23·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{112} \cdot 3^{48}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{112} \cdot 3^{48}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.02128689910\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02128689910\) |
| \(L(\frac{3}{2})\) |
|
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 | \( ( 1 - 2 p T^{2} + 49 T^{4} - 2 p T^{6} - 524 T^{8} - 2 p^{3} T^{10} + 49 p^{4} T^{12} - 2 p^{7} T^{14} + p^{8} T^{16} )^{2} \) |
| 7 | \( ( 1 + 10 T^{2} - 17 T^{4} + 190 T^{6} + 6388 T^{8} + 190 p^{2} T^{10} - 17 p^{4} T^{12} + 10 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 11 | \( ( 1 + 38 T^{2} + 7 p^{2} T^{4} + 13490 T^{6} + 167044 T^{8} + 13490 p^{2} T^{10} + 7 p^{6} T^{12} + 38 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 13 | \( ( 1 + 17 T^{2} + 120 T^{4} + 17 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 17 | \( ( 1 - 16 T^{2} + p^{2} T^{4} )^{8} \) |
| 19 | \( ( 1 + 4 T^{2} - 138 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 23 | \( ( 1 - 74 T^{2} + 3103 T^{4} - 97310 T^{6} + 2454484 T^{8} - 97310 p^{2} T^{10} + 3103 p^{4} T^{12} - 74 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 29 | \( ( 1 - 46 T^{2} + 769 T^{4} + 15410 T^{6} - 672428 T^{8} + 15410 p^{2} T^{10} + 769 p^{4} T^{12} - 46 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 31 | \( ( 1 + 22 T^{2} - 833 T^{4} - 13310 T^{6} + 345844 T^{8} - 13310 p^{2} T^{10} - 833 p^{4} T^{12} + 22 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 37 | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{8} \) |
| 41 | \( ( 1 - 18 T + 199 T^{2} - 1638 T^{3} + 11028 T^{4} - 1638 p T^{5} + 199 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 43 | \( ( 1 - 118 T^{2} + 7231 T^{4} - 353410 T^{6} + 15631972 T^{8} - 353410 p^{2} T^{10} + 7231 p^{4} T^{12} - 118 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 47 | \( ( 1 + 10 T^{2} - 4289 T^{4} - 290 T^{6} + 14162740 T^{8} - 290 p^{2} T^{10} - 4289 p^{4} T^{12} + 10 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 53 | \( ( 1 + 112 T^{2} + 8370 T^{4} + 112 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 59 | \( ( 1 + 218 T^{2} + 28735 T^{4} + 2578286 T^{6} + 175478116 T^{8} + 2578286 p^{2} T^{10} + 28735 p^{4} T^{12} + 218 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 61 | \( ( 1 + 41 T^{2} - 2040 T^{4} + 41 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 67 | \( ( 1 - 106 T^{2} + 5983 T^{4} + 394850 T^{6} - 42995516 T^{8} + 394850 p^{2} T^{10} + 5983 p^{4} T^{12} - 106 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 71 | \( ( 1 - 112 T^{2} + 13002 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 73 | \( ( 1 + 8 T + 156 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{8} \) |
| 79 | \( ( 1 + 118 T^{2} - 1985 T^{4} + 404386 T^{6} + 127246516 T^{8} + 404386 p^{2} T^{10} - 1985 p^{4} T^{12} + 118 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 83 | \( ( 1 + 314 T^{2} + 725 p T^{4} + 7737902 T^{6} + 749487076 T^{8} + 7737902 p^{2} T^{10} + 725 p^{5} T^{12} + 314 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 89 | \( ( 1 - 104 T^{2} + 10770 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 97 | \( ( 1 - 2 T - 41 T^{2} + 298 T^{3} - 7772 T^{4} + 298 p T^{5} - 41 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
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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
−2.10786386696455988344139089192, −1.96208118270159376061571938253, −1.94416650168303929933155522523, −1.91252553831662816066302333607, −1.85084955245004881559271691837, −1.64146383304917320874383614272, −1.57086379368752640300409908761, −1.51260380312304924678703497364, −1.45805277491534659590070024498, −1.41436343390551142203660907249, −1.25045524145073852638542602688, −1.18879300220132645522002042889, −1.18788184378151955144263686771, −1.14254620880565454936334271300, −1.08366499588673027717133682640, −1.06021785133177376723052863279, −0.984944410919786851813581924111, −0.799609868208052794602496520095, −0.796606748988884494736034888806, −0.69004795984492349700182127657, −0.63752596172997659644536155578, −0.25824807202554870178734024550, −0.21444933912082643856655928712, −0.05558751881907821841637125275, −0.02577912477696783202301623509,
0.02577912477696783202301623509, 0.05558751881907821841637125275, 0.21444933912082643856655928712, 0.25824807202554870178734024550, 0.63752596172997659644536155578, 0.69004795984492349700182127657, 0.796606748988884494736034888806, 0.799609868208052794602496520095, 0.984944410919786851813581924111, 1.06021785133177376723052863279, 1.08366499588673027717133682640, 1.14254620880565454936334271300, 1.18788184378151955144263686771, 1.18879300220132645522002042889, 1.25045524145073852638542602688, 1.41436343390551142203660907249, 1.45805277491534659590070024498, 1.51260380312304924678703497364, 1.57086379368752640300409908761, 1.64146383304917320874383614272, 1.85084955245004881559271691837, 1.91252553831662816066302333607, 1.94416650168303929933155522523, 1.96208118270159376061571938253, 2.10786386696455988344139089192
Plot not available for L-functions of degree greater than 10.
