| L(s) = 1 | + 7·16-s + 928·31-s − 1.43e3·61-s − 729·81-s + 5.32e3·121-s + ⋯ |
| L(s) = 1 | + 7/64·16-s + 5.37·31-s − 3.00·61-s − 81-s + 4·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.644548226\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.644548226\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2^2$ | \( 1 + p^{6} T^{4} \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 - 7 T^{4} + p^{12} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 + p^{6} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 + p^{6} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 + 39972098 T^{4} + p^{12} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 13178 T^{2} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 81332062 T^{4} + p^{12} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - 232 T + p^{3} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + p^{6} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 13452309502 T^{4} + p^{12} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 36467062702 T^{4} + p^{12} T^{8} \) |
| 59 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 + 358 T + p^{3} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + p^{6} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + p^{6} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 893662 T^{2} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 433407300622 T^{4} + p^{12} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 + p^{6} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.44088047333363708529169391229, −9.948248138674941177816407463155, −9.678463731623077614938127577386, −9.437699951302257258045107546301, −9.208194089992275265727966730827, −8.563402332066144946439928253125, −8.318120844194051873027789027437, −8.254581961082140733500939770161, −7.977962005503125798969076929768, −7.42097399454339030479681129817, −7.08068238018536699206277874927, −6.84702350477602627744975740930, −6.31323880811286934801440676876, −6.02356370250665433055840566181, −5.97660660924973794957875499798, −5.31562020905777169364986863503, −4.62142724150123569012342760278, −4.54922500887656353602197035628, −4.44357443287396644337596497603, −3.55372622833743811273149660676, −2.92458588307240902277531611796, −2.90394097269245471735939559305, −2.09188174196224252437109491252, −1.26109311214373982530178842783, −0.63488197754918008055775487603,
0.63488197754918008055775487603, 1.26109311214373982530178842783, 2.09188174196224252437109491252, 2.90394097269245471735939559305, 2.92458588307240902277531611796, 3.55372622833743811273149660676, 4.44357443287396644337596497603, 4.54922500887656353602197035628, 4.62142724150123569012342760278, 5.31562020905777169364986863503, 5.97660660924973794957875499798, 6.02356370250665433055840566181, 6.31323880811286934801440676876, 6.84702350477602627744975740930, 7.08068238018536699206277874927, 7.42097399454339030479681129817, 7.977962005503125798969076929768, 8.254581961082140733500939770161, 8.318120844194051873027789027437, 8.563402332066144946439928253125, 9.208194089992275265727966730827, 9.437699951302257258045107546301, 9.678463731623077614938127577386, 9.948248138674941177816407463155, 10.44088047333363708529169391229
