| L(s) = 1 | − 3-s + 8·7-s + 9-s − 4·13-s + 8·19-s − 8·21-s − 6·25-s − 27-s − 8·31-s − 4·37-s + 4·39-s − 8·43-s + 34·49-s − 8·57-s + 12·61-s + 8·63-s − 8·67-s − 12·73-s + 6·75-s − 8·79-s + 81-s − 32·91-s + 8·93-s − 28·97-s + 24·103-s + 28·109-s + 4·111-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 3.02·7-s + 1/3·9-s − 1.10·13-s + 1.83·19-s − 1.74·21-s − 6/5·25-s − 0.192·27-s − 1.43·31-s − 0.657·37-s + 0.640·39-s − 1.21·43-s + 34/7·49-s − 1.05·57-s + 1.53·61-s + 1.00·63-s − 0.977·67-s − 1.40·73-s + 0.692·75-s − 0.900·79-s + 1/9·81-s − 3.35·91-s + 0.829·93-s − 2.84·97-s + 2.36·103-s + 2.68·109-s + 0.379·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27648 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27648 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.354096709\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.354096709\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−10.84849442743057013529606410604, −9.937852275620133476013119550490, −9.826637660192491869357018654726, −8.839451555498636571243071250626, −8.391046533691483698004382219288, −7.81343717072417778826484889358, −7.28654230418818125978122396710, −7.17212008872630953090710242182, −5.64125009969007529682624024748, −5.56983356275681900785379754785, −4.77602747706575131165960838209, −4.57894092460876565195206012945, −3.48999177685615809985045793307, −2.09803110754033007053716721218, −1.46933061420707130494872098274,
1.46933061420707130494872098274, 2.09803110754033007053716721218, 3.48999177685615809985045793307, 4.57894092460876565195206012945, 4.77602747706575131165960838209, 5.56983356275681900785379754785, 5.64125009969007529682624024748, 7.17212008872630953090710242182, 7.28654230418818125978122396710, 7.81343717072417778826484889358, 8.391046533691483698004382219288, 8.839451555498636571243071250626, 9.826637660192491869357018654726, 9.937852275620133476013119550490, 10.84849442743057013529606410604
