| L(s) = 1 | + 48·19-s − 4·25-s − 32·49-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 + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯ |
| L(s) = 1 | + 11.0·19-s − 4/5·25-s − 4.57·49-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 + 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 + 0.0646·239-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.312079449\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.312079449\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| good | 7 | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - 6 T + p T^{2} )^{8} \) |
| 23 | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - 64 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 - 62 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 - 116 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 10 T + p T^{2} )^{4}( 1 + 10 T + p T^{2} )^{4} \) |
| 67 | \( ( 1 + 82 T^{2} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 14 T + p T^{2} )^{4}( 1 + 14 T + p T^{2} )^{4} \) |
| 79 | \( ( 1 - 154 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 - 160 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 170 T^{2} + p^{2} T^{4} )^{4} \) |
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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
−3.44724091023850918827552373022, −3.39691999398368832798804146311, −3.28461533435626023917218306903, −3.15365691154363136101044046983, −3.15028409367121511870837872644, −3.09749763107345488331261833152, −3.09707335075820174095995405000, −3.06994075515037052393463981928, −2.88116437707920679438528231577, −2.68918879805933687814784364726, −2.40668547404895229682789301321, −2.32355084134231624538487661563, −2.16128805336860579397448749881, −1.83985426438235469516644842435, −1.77893420311579428236189294681, −1.72537053007596836003826237594, −1.68236062147146243018251682358, −1.37962018562395288118061553738, −1.16297142327644936802119569516, −1.07386363188978732537941597102, −0.994539979741206238024178999244, −0.854863527172030075810063330618, −0.806219608053476865036196701292, −0.51404573728447672218801146860, −0.12977991383724321686711533907,
0.12977991383724321686711533907, 0.51404573728447672218801146860, 0.806219608053476865036196701292, 0.854863527172030075810063330618, 0.994539979741206238024178999244, 1.07386363188978732537941597102, 1.16297142327644936802119569516, 1.37962018562395288118061553738, 1.68236062147146243018251682358, 1.72537053007596836003826237594, 1.77893420311579428236189294681, 1.83985426438235469516644842435, 2.16128805336860579397448749881, 2.32355084134231624538487661563, 2.40668547404895229682789301321, 2.68918879805933687814784364726, 2.88116437707920679438528231577, 3.06994075515037052393463981928, 3.09707335075820174095995405000, 3.09749763107345488331261833152, 3.15028409367121511870837872644, 3.15365691154363136101044046983, 3.28461533435626023917218306903, 3.39691999398368832798804146311, 3.44724091023850918827552373022
Plot not available for L-functions of degree greater than 10.
