| L(s) = 1 | − 7-s + 13-s + 19-s + 2·25-s + 31-s + 37-s + 43-s − 2·61-s + 67-s + 73-s + 79-s − 91-s − 2·97-s − 2·103-s + 109-s + 2·121-s + 127-s + 131-s − 133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + ⋯ |
| L(s) = 1 | − 7-s + 13-s + 19-s + 2·25-s + 31-s + 37-s + 43-s − 2·61-s + 67-s + 73-s + 79-s − 91-s − 2·97-s − 2·103-s + 109-s + 2·121-s + 127-s + 131-s − 133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.319614632\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.319614632\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
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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
−9.317780582646917971321759921381, −9.237170542814613976888280239677, −8.512769220661477125424529172716, −8.372166272236905830619959097365, −7.898944490705169190623942359637, −7.47935974545566306832226860597, −6.88843538891503310155864051112, −6.81285161015878071365300197384, −6.19318321187977936776921390043, −6.06835495951443995566578729167, −5.49192022219242850871640485841, −5.06999682094199822997274891697, −4.53395818778021085677674268190, −4.22654243073684989423628408290, −3.46109840750289883802906301299, −3.31613638099334255020726173121, −2.75592969845313402678990414357, −2.35650938725235909753171928929, −1.30718116133893840645938333017, −0.928506364829398284787484866633,
0.928506364829398284787484866633, 1.30718116133893840645938333017, 2.35650938725235909753171928929, 2.75592969845313402678990414357, 3.31613638099334255020726173121, 3.46109840750289883802906301299, 4.22654243073684989423628408290, 4.53395818778021085677674268190, 5.06999682094199822997274891697, 5.49192022219242850871640485841, 6.06835495951443995566578729167, 6.19318321187977936776921390043, 6.81285161015878071365300197384, 6.88843538891503310155864051112, 7.47935974545566306832226860597, 7.898944490705169190623942359637, 8.372166272236905830619959097365, 8.512769220661477125424529172716, 9.237170542814613976888280239677, 9.317780582646917971321759921381
