| L(s) = 1 | − 3-s − 3·5-s + 9-s + 2·13-s + 3·15-s + 4·25-s − 27-s + 13·31-s + 2·37-s − 2·39-s + 14·43-s − 3·45-s + 5·49-s − 3·53-s − 6·65-s − 16·67-s − 4·75-s − 2·79-s + 81-s − 21·83-s + 6·89-s − 13·93-s − 2·111-s + 2·117-s + 5·121-s + 3·125-s + 127-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s + 1/3·9-s + 0.554·13-s + 0.774·15-s + 4/5·25-s − 0.192·27-s + 2.33·31-s + 0.328·37-s − 0.320·39-s + 2.13·43-s − 0.447·45-s + 5/7·49-s − 0.412·53-s − 0.744·65-s − 1.95·67-s − 0.461·75-s − 0.225·79-s + 1/9·81-s − 2.30·83-s + 0.635·89-s − 1.34·93-s − 0.189·111-s + 0.184·117-s + 5/11·121-s + 0.268·125-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.060402985\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.060402985\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 5 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 83 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 109 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 9 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 49 T^{2} + p^{2} T^{4} \) |
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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
−8.564648533240610310549269103095, −7.942100412887309597006632095108, −7.58200715680681862168331404168, −7.25079973539148881724492197468, −6.68160639247936357805582314704, −6.08666617113268198857423296535, −5.91551359554502060335631050066, −5.15822889542356211269258969245, −4.49426594769199031754778969635, −4.30116304932395186285750753439, −3.79252808342293966002072243019, −3.03082081147126748997055090277, −2.56604451964154623411663464362, −1.38621326405113749966545486235, −0.61122257091611540630596438243,
0.61122257091611540630596438243, 1.38621326405113749966545486235, 2.56604451964154623411663464362, 3.03082081147126748997055090277, 3.79252808342293966002072243019, 4.30116304932395186285750753439, 4.49426594769199031754778969635, 5.15822889542356211269258969245, 5.91551359554502060335631050066, 6.08666617113268198857423296535, 6.68160639247936357805582314704, 7.25079973539148881724492197468, 7.58200715680681862168331404168, 7.942100412887309597006632095108, 8.564648533240610310549269103095
