| L(s) = 1 | + 12·11-s + 16-s + 32·31-s + 84·41-s + 16·61-s + 48·101-s + 34·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 12·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | + 3.61·11-s + 1/4·16-s + 5.74·31-s + 13.1·41-s + 2.04·61-s + 4.77·101-s + 3.09·121-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 + 0.0760·173-s + 0.904·176-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(27.08066517\) |
| \(L(\frac12)\) |
\(\approx\) |
\(27.08066517\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T^{4} + T^{8} \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 + 23 T^{4} + p^{4} T^{8} )( 1 + 71 T^{4} + p^{4} T^{8} ) \) |
| 11 | \( ( 1 - 3 T + 14 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{4} \) |
| 13 | \( 1 + 142 T^{4} - 8397 T^{8} + 142 p^{4} T^{12} + p^{8} T^{16} \) |
| 17 | \( ( 1 + 47 T^{4} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( 1 + 958 T^{4} + 637923 T^{8} + 958 p^{4} T^{12} + p^{8} T^{16} \) |
| 29 | \( ( 1 - 46 T^{2} + 1275 T^{4} - 46 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - 2062 T^{4} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 21 T + 188 T^{2} - 21 p T^{3} + p^{2} T^{4} )^{4} \) |
| 43 | \( 1 + 217 T^{4} - 3371712 T^{8} + 217 p^{4} T^{12} + p^{8} T^{16} \) |
| 47 | \( 1 + 1054 T^{4} - 3768765 T^{8} + 1054 p^{4} T^{12} + p^{8} T^{16} \) |
| 53 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 + 29 T^{2} - 2640 T^{4} + 29 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( 1 - 8183 T^{4} + 46810368 T^{8} - 8183 p^{4} T^{12} + p^{8} T^{16} \) |
| 71 | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 5617 T^{4} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{2}( 1 + 131 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | \( 1 + 13294 T^{4} + 129272115 T^{8} + 13294 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( ( 1 + 130 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( 1 + 4657 T^{4} - 66841632 T^{8} + 4657 p^{4} T^{12} + p^{8} T^{16} \) |
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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
−4.25676001045675157739447144822, −3.98059672559152549664789549557, −3.77514477809048064602576932030, −3.74736576765496857301960795222, −3.61366921042722631355712365391, −3.47498959117995731945360308689, −3.37959703726181684898633661725, −3.35768836379936555828180629580, −2.98435073269791264841752959782, −2.66590080420127177562767488540, −2.61544267416425559663688553049, −2.56016403842481588775586152513, −2.46478719405779264742020312495, −2.46016761592660748385908540592, −2.36193233733325809696846817832, −2.16614880284205199443177110876, −2.02292298654385354391132159517, −1.44304911140765108753078252558, −1.27246313750697689673343720515, −1.12493546108315931046033312760, −1.06534368102974631754345174826, −1.03038053774391207503686253563, −0.976686138621032752247737053073, −0.76489559445784574508932143395, −0.39267895150733769667156888729,
0.39267895150733769667156888729, 0.76489559445784574508932143395, 0.976686138621032752247737053073, 1.03038053774391207503686253563, 1.06534368102974631754345174826, 1.12493546108315931046033312760, 1.27246313750697689673343720515, 1.44304911140765108753078252558, 2.02292298654385354391132159517, 2.16614880284205199443177110876, 2.36193233733325809696846817832, 2.46016761592660748385908540592, 2.46478719405779264742020312495, 2.56016403842481588775586152513, 2.61544267416425559663688553049, 2.66590080420127177562767488540, 2.98435073269791264841752959782, 3.35768836379936555828180629580, 3.37959703726181684898633661725, 3.47498959117995731945360308689, 3.61366921042722631355712365391, 3.74736576765496857301960795222, 3.77514477809048064602576932030, 3.98059672559152549664789549557, 4.25676001045675157739447144822
Plot not available for L-functions of degree greater than 10.
