| L(s) = 1 | + 2·4-s − 8·7-s − 38·13-s − 12·16-s − 2·19-s + 44·25-s − 16·28-s − 2·31-s + 88·37-s + 10·43-s − 50·49-s − 76·52-s + 106·61-s − 56·64-s + 94·67-s − 230·73-s − 4·76-s + 298·79-s + 304·91-s + 46·97-s + 88·100-s + 58·103-s + 106·109-s + 96·112-s − 52·121-s − 4·124-s + 127-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 8/7·7-s − 2.92·13-s − 3/4·16-s − 0.105·19-s + 1.75·25-s − 4/7·28-s − 0.0645·31-s + 2.37·37-s + 0.232·43-s − 1.02·49-s − 1.46·52-s + 1.73·61-s − 7/8·64-s + 1.40·67-s − 3.15·73-s − 0.0526·76-s + 3.77·79-s + 3.34·91-s + 0.474·97-s + 0.879·100-s + 0.563·103-s + 0.972·109-s + 6/7·112-s − 0.429·121-s − 0.0322·124-s + 0.00787·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.191461866\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.191461866\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{4} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 44 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 52 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 19 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 92 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 332 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 718 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 44 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 2978 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 2018 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 5564 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6956 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 53 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 47 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 3548 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 115 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 149 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 628 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 14978 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 23 T + p^{2} T^{2} )^{2} \) |
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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
−12.54455993358677483039086422219, −11.59780704668624355827121509267, −11.28471511667498364251942295329, −10.65580591417550802998070428772, −10.06135845986083474848106245241, −9.729644875143522694314980561422, −9.414100029546866146886589050610, −8.865524501087664119772812552398, −8.114096765174250420147438059714, −7.35141973747284918700562622744, −7.30632926900892380082800518647, −6.45526979117190614881462379167, −6.39582510764647680659024910873, −5.30578541684810471771256745464, −4.83813165467723230121804949837, −4.33086028165607381785927110124, −3.24224816853883082575395540320, −2.63511673565408497016350189676, −2.23270220322015391461781691577, −0.54786383871887702679038124847,
0.54786383871887702679038124847, 2.23270220322015391461781691577, 2.63511673565408497016350189676, 3.24224816853883082575395540320, 4.33086028165607381785927110124, 4.83813165467723230121804949837, 5.30578541684810471771256745464, 6.39582510764647680659024910873, 6.45526979117190614881462379167, 7.30632926900892380082800518647, 7.35141973747284918700562622744, 8.114096765174250420147438059714, 8.865524501087664119772812552398, 9.414100029546866146886589050610, 9.729644875143522694314980561422, 10.06135845986083474848106245241, 10.65580591417550802998070428772, 11.28471511667498364251942295329, 11.59780704668624355827121509267, 12.54455993358677483039086422219
