| L(s) = 1 | + 2·3-s + 7-s + 9-s − 6·11-s − 6·13-s − 4·16-s + 2·19-s + 2·21-s + 3·25-s − 4·27-s − 6·31-s − 12·33-s − 12·39-s + 6·41-s − 6·43-s − 8·48-s − 6·49-s + 4·57-s − 6·59-s + 63-s + 6·75-s − 6·77-s − 11·81-s + 18·89-s − 6·91-s − 12·93-s − 2·97-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.377·7-s + 1/3·9-s − 1.80·11-s − 1.66·13-s − 16-s + 0.458·19-s + 0.436·21-s + 3/5·25-s − 0.769·27-s − 1.07·31-s − 2.08·33-s − 1.92·39-s + 0.937·41-s − 0.914·43-s − 1.15·48-s − 6/7·49-s + 0.529·57-s − 0.781·59-s + 0.125·63-s + 0.692·75-s − 0.683·77-s − 1.22·81-s + 1.90·89-s − 0.628·91-s − 1.24·93-s − 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 159201 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 159201 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| 19 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 49 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−9.039979997014562825044265080554, −8.502603613321408717817349631996, −7.913940031173972780990697013470, −7.71443333380264699252993731176, −7.28656108977016870735590509908, −6.78060270001303809445259308973, −5.95916257479035645844689946425, −5.21308452939277172914953637630, −5.00451322451101784358612240250, −4.43529178939998421784590859097, −3.55119326228708525991147291262, −2.85047031852211250651651137569, −2.49336049098788615823286217953, −1.87615766237270307681991770414, 0,
1.87615766237270307681991770414, 2.49336049098788615823286217953, 2.85047031852211250651651137569, 3.55119326228708525991147291262, 4.43529178939998421784590859097, 5.00451322451101784358612240250, 5.21308452939277172914953637630, 5.95916257479035645844689946425, 6.78060270001303809445259308973, 7.28656108977016870735590509908, 7.71443333380264699252993731176, 7.913940031173972780990697013470, 8.502603613321408717817349631996, 9.039979997014562825044265080554
