| L(s) = 1 | − 3·3-s − 2·7-s + 6·9-s + 3·13-s + 8·19-s + 6·21-s − 5·25-s − 9·27-s − 9·39-s + 5·43-s − 11·49-s − 24·57-s − 13·61-s − 12·63-s + 27·67-s + 7·73-s + 15·75-s − 21·79-s + 9·81-s − 6·91-s + 24·97-s + 36·109-s + 18·117-s + 22·121-s + 127-s − 15·129-s + 131-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.755·7-s + 2·9-s + 0.832·13-s + 1.83·19-s + 1.30·21-s − 25-s − 1.73·27-s − 1.44·39-s + 0.762·43-s − 1.57·49-s − 3.17·57-s − 1.66·61-s − 1.51·63-s + 3.29·67-s + 0.819·73-s + 1.73·75-s − 2.36·79-s + 81-s − 0.628·91-s + 2.43·97-s + 3.44·109-s + 1.66·117-s + 2·121-s + 0.0887·127-s − 1.32·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8783401229\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8783401229\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 19 | $C_2$ | \( 1 - 8 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 11 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
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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
−10.13897162808116930670349604205, −10.00311657342454386641840395283, −9.736228114929307421622786169791, −9.121033248371343188949767000476, −8.807284673552640923989318948932, −8.004075526395432552469405186401, −7.67801626416124460551599245233, −7.26394257254266819427114318211, −6.71175807179879474444745423776, −6.37311003893507907746973752706, −5.94013360447493335798086731790, −5.65117602599111564984811374913, −5.11273958969276149091409984587, −4.71747194763352177561202472590, −4.08046024924639346782630873004, −3.48685352687846023996732429693, −3.13379200516269274789181400945, −2.06899686290975617007166025818, −1.27241638092739712913031359346, −0.55974054361500708227279987436,
0.55974054361500708227279987436, 1.27241638092739712913031359346, 2.06899686290975617007166025818, 3.13379200516269274789181400945, 3.48685352687846023996732429693, 4.08046024924639346782630873004, 4.71747194763352177561202472590, 5.11273958969276149091409984587, 5.65117602599111564984811374913, 5.94013360447493335798086731790, 6.37311003893507907746973752706, 6.71175807179879474444745423776, 7.26394257254266819427114318211, 7.67801626416124460551599245233, 8.004075526395432552469405186401, 8.807284673552640923989318948932, 9.121033248371343188949767000476, 9.736228114929307421622786169791, 10.00311657342454386641840395283, 10.13897162808116930670349604205
