L(s) = 1 | − 2·3-s − 2·7-s + 9-s − 8·13-s − 12·19-s + 4·21-s − 10·25-s + 4·27-s − 8·31-s + 20·37-s + 16·39-s + 8·43-s + 3·49-s + 24·57-s − 16·61-s − 2·63-s − 16·67-s − 12·73-s + 20·75-s − 32·79-s − 11·81-s + 16·91-s + 16·93-s − 4·97-s − 8·103-s + 20·109-s − 40·111-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 0.755·7-s + 1/3·9-s − 2.21·13-s − 2.75·19-s + 0.872·21-s − 2·25-s + 0.769·27-s − 1.43·31-s + 3.28·37-s + 2.56·39-s + 1.21·43-s + 3/7·49-s + 3.17·57-s − 2.04·61-s − 0.251·63-s − 1.95·67-s − 1.40·73-s + 2.30·75-s − 3.60·79-s − 1.22·81-s + 1.67·91-s + 1.65·93-s − 0.406·97-s − 0.788·103-s + 1.91·109-s − 3.79·111-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 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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
−7.85643204119898614530616966999, −7.56688031432348988896561867945, −7.31000008746359385531868028906, −6.52662751327895896284855360684, −6.17236708003451018025878030760, −5.88296122377187582734081048905, −5.48532390305501388420276275059, −4.57048039337710861640747663017, −4.39654298826148920136877245563, −3.99709320456544753258230397198, −2.70117202411251547524653201529, −2.62612435759905589929421296030, −1.69348648530400294148534200114, 0, 0,
1.69348648530400294148534200114, 2.62612435759905589929421296030, 2.70117202411251547524653201529, 3.99709320456544753258230397198, 4.39654298826148920136877245563, 4.57048039337710861640747663017, 5.48532390305501388420276275059, 5.88296122377187582734081048905, 6.17236708003451018025878030760, 6.52662751327895896284855360684, 7.31000008746359385531868028906, 7.56688031432348988896561867945, 7.85643204119898614530616966999