| L(s) = 1 | − 440·17-s + 2.21e3·25-s − 5.12e3·41-s + 1.82e3·49-s − 1.26e4·73-s − 6.20e3·89-s − 3.20e4·97-s − 4.31e4·113-s − 5.77e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2.51e4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 1.52·17-s + 3.53·25-s − 3.05·41-s + 0.761·49-s − 2.37·73-s − 0.782·89-s − 3.40·97-s − 3.37·113-s − 3.94·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s − 0.880·169-s + 3.34e−5·173-s + 3.12e−5·179-s + 3.05e−5·181-s + 2.74e−5·191-s + 2.68e−5·193-s + 2.57e−5·197-s + 2.52e−5·199-s + 2.24e−5·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.3524653273\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3524653273\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 1106 T^{2} + p^{8} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 914 T^{2} + p^{8} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 28850 T^{2} + p^{8} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 12574 T^{2} + p^{8} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 110 T + p^{4} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + 249842 T^{2} + p^{8} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 544130 T^{2} + p^{8} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 1362578 T^{2} + p^{8} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 209710 T^{2} + p^{8} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 1810658 T^{2} + p^{8} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 1282 T + p^{4} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 512690 T^{2} + p^{8} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 2900930 T^{2} + p^{8} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 4469038 T^{2} + p^{8} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 15952078 T^{2} + p^{8} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 26770082 T^{2} + p^{8} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 29653874 T^{2} + p^{8} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 13483010 T^{2} + p^{8} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 3170 T + p^{4} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 75470162 T^{2} + p^{8} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 75317234 T^{2} + p^{8} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 1550 T + p^{4} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 8018 T + p^{4} T^{2} )^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.47558550438694229661360652699, −6.39928800551395957533691667859, −6.16947464690023952215529571973, −5.52536630394091331392795301067, −5.42828868954613050805984154433, −5.29716205094910300135279392694, −5.28856908412821609878728884023, −4.66605440763064075095509648762, −4.65564467741065907465522177895, −4.38501336170579193515646339830, −4.34419432639281575875698358882, −3.81230511651986038614182245396, −3.51686197890054678591822252265, −3.50478476595453762539977261569, −3.08164465302307228664890810425, −2.76454794233027368288007727961, −2.57639555351598706784596124319, −2.31284007937224524976300177698, −2.28728801237496473850217718334, −1.46911031922182331534834956634, −1.32479266077671511772195044568, −1.20601036999005065218697784645, −1.11005164013444104405634741215, −0.16693581597358472653972015211, −0.16340167187165227893247802036,
0.16340167187165227893247802036, 0.16693581597358472653972015211, 1.11005164013444104405634741215, 1.20601036999005065218697784645, 1.32479266077671511772195044568, 1.46911031922182331534834956634, 2.28728801237496473850217718334, 2.31284007937224524976300177698, 2.57639555351598706784596124319, 2.76454794233027368288007727961, 3.08164465302307228664890810425, 3.50478476595453762539977261569, 3.51686197890054678591822252265, 3.81230511651986038614182245396, 4.34419432639281575875698358882, 4.38501336170579193515646339830, 4.65564467741065907465522177895, 4.66605440763064075095509648762, 5.28856908412821609878728884023, 5.29716205094910300135279392694, 5.42828868954613050805984154433, 5.52536630394091331392795301067, 6.16947464690023952215529571973, 6.39928800551395957533691667859, 6.47558550438694229661360652699
