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
−8.142737265603496815088661734936, −7.64230674264070832489473958476, −7.56688031432348988896561867945, −7.03619200467257105345639081242, −6.52662751327895896284855360684, −5.72643285643591914021095909045, −5.48532390305501388420276275059, −5.20704742703837539203927825545, −4.57048039337710861640747663017, −4.38921941756740989472338293857, −3.09009811816266066443914661488, −2.62612435759905589929421296030, −2.38160133368819946421648201021, −0.867770609357001154731944493797, 0,
0.867770609357001154731944493797, 2.38160133368819946421648201021, 2.62612435759905589929421296030, 3.09009811816266066443914661488, 4.38921941756740989472338293857, 4.57048039337710861640747663017, 5.20704742703837539203927825545, 5.48532390305501388420276275059, 5.72643285643591914021095909045, 6.52662751327895896284855360684, 7.03619200467257105345639081242, 7.56688031432348988896561867945, 7.64230674264070832489473958476, 8.142737265603496815088661734936