| L(s) = 1 | + 2·3-s + 9-s + 4·11-s + 4·13-s − 8·23-s − 2·25-s − 4·27-s + 8·33-s + 4·37-s + 8·39-s − 2·49-s − 4·59-s + 20·61-s − 16·69-s − 8·71-s + 4·73-s − 4·75-s − 11·81-s − 20·83-s − 4·97-s + 4·99-s − 4·107-s − 12·109-s + 8·111-s + 4·117-s + 6·121-s + 127-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s + 1.20·11-s + 1.10·13-s − 1.66·23-s − 2/5·25-s − 0.769·27-s + 1.39·33-s + 0.657·37-s + 1.28·39-s − 2/7·49-s − 0.520·59-s + 2.56·61-s − 1.92·69-s − 0.949·71-s + 0.468·73-s − 0.461·75-s − 1.22·81-s − 2.19·83-s − 0.406·97-s + 0.402·99-s − 0.386·107-s − 1.14·109-s + 0.759·111-s + 0.369·117-s + 6/11·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.964811619\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.964811619\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−10.03045773630582112349040341124, −9.822353057501042439003867593911, −9.248505727642093065314855285982, −8.688757670023775560985236864470, −8.348781260172839309711754415932, −7.912976180298523208679694953657, −7.22622360838775834162972537900, −6.56751042591285918771393023231, −6.01787329173103130344580621766, −5.51068329881977020855931527274, −4.29545114391794679641863097735, −3.94357278164872657753927112929, −3.34650501351684819915646708153, −2.39828951388922368151652610487, −1.50758218448160494122558467671,
1.50758218448160494122558467671, 2.39828951388922368151652610487, 3.34650501351684819915646708153, 3.94357278164872657753927112929, 4.29545114391794679641863097735, 5.51068329881977020855931527274, 6.01787329173103130344580621766, 6.56751042591285918771393023231, 7.22622360838775834162972537900, 7.912976180298523208679694953657, 8.348781260172839309711754415932, 8.688757670023775560985236864470, 9.248505727642093065314855285982, 9.822353057501042439003867593911, 10.03045773630582112349040341124
