| L(s) = 1 | + 6·3-s + 52·7-s − 45·9-s − 100·13-s + 716·19-s + 312·21-s + 962·25-s − 756·27-s − 1.48e3·31-s − 3.74e3·37-s − 600·39-s + 524·43-s − 2.77e3·49-s + 4.29e3·57-s + 2.97e3·61-s − 2.34e3·63-s + 8.97e3·67-s + 580·73-s + 5.77e3·75-s + 1.96e4·79-s − 891·81-s − 5.20e3·91-s − 8.90e3·93-s − 956·97-s + 4.27e3·103-s + 9.50e3·109-s − 2.24e4·111-s + ⋯ |
| L(s) = 1 | + 2/3·3-s + 1.06·7-s − 5/9·9-s − 0.591·13-s + 1.98·19-s + 0.707·21-s + 1.53·25-s − 1.03·27-s − 1.54·31-s − 2.73·37-s − 0.394·39-s + 0.283·43-s − 1.15·49-s + 1.32·57-s + 0.798·61-s − 0.589·63-s + 1.99·67-s + 0.108·73-s + 1.02·75-s + 3.14·79-s − 0.135·81-s − 0.627·91-s − 1.02·93-s − 0.101·97-s + 0.403·103-s + 0.799·109-s − 1.82·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(3.306409577\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.306409577\) |
| \(L(3)\) |
|
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 | $C_2$ | \( 1 - 2 p T + p^{4} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 962 T^{2} + p^{8} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 26 T + p^{4} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 15170 T^{2} + p^{8} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 50 T + p^{4} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 125570 T^{2} + p^{8} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 358 T + p^{4} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 420290 T^{2} + p^{8} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 666238 T^{2} + p^{8} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 742 T + p^{4} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 1874 T + p^{4} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 155710 T^{2} + p^{8} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 262 T + p^{4} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 6879362 T^{2} + p^{8} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 15571010 T^{2} + p^{8} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 20937410 T^{2} + p^{8} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 1486 T + p^{4} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4486 T + p^{4} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 38122562 T^{2} + p^{8} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 290 T + p^{4} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 9818 T + p^{4} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 44355074 T^{2} + p^{8} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 64012610 T^{2} + p^{8} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 478 T + p^{4} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.13572084323587795842806660455, −11.59214694489718503583264401374, −11.06977077591468786882603903159, −10.82882629240916553376013116433, −10.07486325947133788879535089804, −9.426504158342018505122057424575, −9.209624262902816517874453476204, −8.340427143598179235707095022318, −8.339522437574098527424032689293, −7.45126370185651206957102741019, −7.20920406904615737950802354198, −6.51861008995150278593856158806, −5.35627037593822518407689628404, −5.31960352023633933881650303822, −4.72280031133293900194793702409, −3.42911193892990786501886081777, −3.40291170043334504149268685203, −2.29090268279638721015791439585, −1.66263590166931587735789856654, −0.63398851782687898838675883218,
0.63398851782687898838675883218, 1.66263590166931587735789856654, 2.29090268279638721015791439585, 3.40291170043334504149268685203, 3.42911193892990786501886081777, 4.72280031133293900194793702409, 5.31960352023633933881650303822, 5.35627037593822518407689628404, 6.51861008995150278593856158806, 7.20920406904615737950802354198, 7.45126370185651206957102741019, 8.339522437574098527424032689293, 8.340427143598179235707095022318, 9.209624262902816517874453476204, 9.426504158342018505122057424575, 10.07486325947133788879535089804, 10.82882629240916553376013116433, 11.06977077591468786882603903159, 11.59214694489718503583264401374, 12.13572084323587795842806660455
