L(s) = 1 | + 2·3-s + 9-s + 8·11-s − 8·13-s − 16·23-s − 10·25-s − 4·27-s + 16·33-s + 20·37-s − 16·39-s − 8·47-s + 49-s − 20·59-s − 16·61-s − 32·69-s − 12·73-s − 20·75-s − 11·81-s − 4·83-s − 4·97-s + 8·99-s + 32·107-s + 20·109-s + 40·111-s − 8·117-s + 26·121-s + 127-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 1/3·9-s + 2.41·11-s − 2.21·13-s − 3.33·23-s − 2·25-s − 0.769·27-s + 2.78·33-s + 3.28·37-s − 2.56·39-s − 1.16·47-s + 1/7·49-s − 2.60·59-s − 2.04·61-s − 3.85·69-s − 1.40·73-s − 2.30·75-s − 1.22·81-s − 0.439·83-s − 0.406·97-s + 0.804·99-s + 3.09·107-s + 1.91·109-s + 3.79·111-s − 0.739·117-s + 2.36·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−8.247827016681358338699106327510, −7.85643204119898614530616966999, −7.53632864653844024945923131798, −7.31000008746359385531868028906, −6.19321040018157050389604631782, −6.17236708003451018025878030760, −5.88296122377187582734081048905, −4.62285613804853643208241606946, −4.39654298826148920136877245563, −3.99709320456544753258230397198, −3.39651452170084757716747861999, −2.70117202411251547524653201529, −2.00516068697428410265218115685, −1.69348648530400294148534200114, 0,
1.69348648530400294148534200114, 2.00516068697428410265218115685, 2.70117202411251547524653201529, 3.39651452170084757716747861999, 3.99709320456544753258230397198, 4.39654298826148920136877245563, 4.62285613804853643208241606946, 5.88296122377187582734081048905, 6.17236708003451018025878030760, 6.19321040018157050389604631782, 7.31000008746359385531868028906, 7.53632864653844024945923131798, 7.85643204119898614530616966999, 8.247827016681358338699106327510