| L(s) = 1 | + 2·3-s − 6·5-s + 3·9-s + 6·11-s − 12·15-s + 2·19-s + 6·23-s + 19·25-s + 10·27-s + 12·29-s − 8·31-s + 12·33-s + 2·37-s − 18·45-s − 6·53-s − 36·55-s + 4·57-s + 6·59-s + 6·61-s + 6·67-s + 12·69-s − 12·73-s + 38·75-s − 6·79-s + 20·81-s + 12·83-s + 24·87-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 2.68·5-s + 9-s + 1.80·11-s − 3.09·15-s + 0.458·19-s + 1.25·23-s + 19/5·25-s + 1.92·27-s + 2.22·29-s − 1.43·31-s + 2.08·33-s + 0.328·37-s − 2.68·45-s − 0.824·53-s − 4.85·55-s + 0.529·57-s + 0.781·59-s + 0.768·61-s + 0.733·67-s + 1.44·69-s − 1.40·73-s + 4.38·75-s − 0.675·79-s + 20/9·81-s + 1.31·83-s + 2.57·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.171616511\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.171616511\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 6 T + 73 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 12 T + 121 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 6 T + 91 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 12 T + 137 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−10.51081854527704736715709239042, −10.10998531859165168631929583354, −9.297607396541808730424968762583, −9.226783850815443263566038497664, −8.622022619969845210406207103158, −8.439237015467873143160342965635, −7.958483362041086165602091025108, −7.68241272580544848151433283162, −6.92927672538402624308263521748, −6.92251881430001579233400816378, −6.56873457262378479271924286932, −5.52663103598904008778325998195, −4.72113833715654742246823404797, −4.51326230884320034009920753224, −3.83498914311003899237019291344, −3.76917325712840530968217257658, −3.05901214614047173021603400116, −2.75280751449364011024900100461, −1.42204531492908734736656466423, −0.793224935602783770073573603726,
0.793224935602783770073573603726, 1.42204531492908734736656466423, 2.75280751449364011024900100461, 3.05901214614047173021603400116, 3.76917325712840530968217257658, 3.83498914311003899237019291344, 4.51326230884320034009920753224, 4.72113833715654742246823404797, 5.52663103598904008778325998195, 6.56873457262378479271924286932, 6.92251881430001579233400816378, 6.92927672538402624308263521748, 7.68241272580544848151433283162, 7.958483362041086165602091025108, 8.439237015467873143160342965635, 8.622022619969845210406207103158, 9.226783850815443263566038497664, 9.297607396541808730424968762583, 10.10998531859165168631929583354, 10.51081854527704736715709239042
