| L(s) = 1 | + 8·9-s + 32·19-s + 48·49-s + 20·81-s + 80·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 80·169-s + 256·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
| L(s) = 1 | + 8/3·9-s + 7.34·19-s + 48/7·49-s + 20/9·81-s + 7.27·121-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 + 6.15·169-s + 19.5·171-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(48.53659732\) |
| \(L(\frac12)\) |
\(\approx\) |
\(48.53659732\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + p^{4} T^{8} )^{2} \) |
| 3 | \( ( 1 - 4 T^{2} + 14 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 - 12 T^{2} + 18 p T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 11 | \( ( 1 - 20 T^{2} + 310 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 13 | \( ( 1 - 20 T^{2} + 406 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 17 | \( ( 1 - 52 T^{2} + 1222 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 19 | \( ( 1 - 4 T + 34 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{8} \) |
| 23 | \( ( 1 + 84 T^{2} + 2814 T^{4} + 84 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 29 | \( ( 1 + 76 T^{2} + 2838 T^{4} + 76 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 31 | \( ( 1 - 76 T^{2} + 2854 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 37 | \( ( 1 - 84 T^{2} + 3702 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 41 | \( ( 1 - 84 T^{2} + 3974 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 43 | \( ( 1 + 60 T^{2} + 4206 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 47 | \( ( 1 + 180 T^{2} + 12510 T^{4} + 180 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 53 | \( ( 1 + 68 T^{2} + 4182 T^{4} + 68 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 59 | \( ( 1 - 212 T^{2} + 18166 T^{4} - 212 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 61 | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{8} \) |
| 67 | \( ( 1 + 252 T^{2} + 24846 T^{4} + 252 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 71 | \( ( 1 + 92 T^{2} + 10150 T^{4} + 92 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 73 | \( ( 1 + 228 T^{2} + 23526 T^{4} + 228 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 79 | \( ( 1 - 44 T^{2} - 5466 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 83 | \( ( 1 - 196 T^{2} + 22990 T^{4} - 196 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 89 | \( ( 1 - 260 T^{2} + 32230 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 97 | \( ( 1 + 260 T^{2} + 32518 T^{4} + 260 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
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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
−2.96441550796902550393936071887, −2.94246947219388449761274531185, −2.79817797195833232287349435283, −2.64264435077479216324767499882, −2.42320260286703554364015570929, −2.36632448254851699456134771236, −2.36254722579250332276960376991, −2.35403516505530799962645786220, −2.25402484736388231680115844041, −2.06002322414140812073424266669, −1.89323932960824113722280570522, −1.85620693821595168788650718414, −1.80382590912374269806675394530, −1.61463295971737360414933234567, −1.53250749395374940248601137918, −1.50535337697377658415259683086, −1.43819612774509355104409979434, −1.08715362494179737449356831718, −1.04439724721712428308399753087, −0.971536533219536437717074305587, −0.941185867691920606161999502970, −0.851762746847184339405450879918, −0.75180331196397507009550300455, −0.58758149524317795814131619814, −0.30530277897190286287322627838,
0.30530277897190286287322627838, 0.58758149524317795814131619814, 0.75180331196397507009550300455, 0.851762746847184339405450879918, 0.941185867691920606161999502970, 0.971536533219536437717074305587, 1.04439724721712428308399753087, 1.08715362494179737449356831718, 1.43819612774509355104409979434, 1.50535337697377658415259683086, 1.53250749395374940248601137918, 1.61463295971737360414933234567, 1.80382590912374269806675394530, 1.85620693821595168788650718414, 1.89323932960824113722280570522, 2.06002322414140812073424266669, 2.25402484736388231680115844041, 2.35403516505530799962645786220, 2.36254722579250332276960376991, 2.36632448254851699456134771236, 2.42320260286703554364015570929, 2.64264435077479216324767499882, 2.79817797195833232287349435283, 2.94246947219388449761274531185, 2.96441550796902550393936071887
Plot not available for L-functions of degree greater than 10.
