| L(s) = 1 | + 2·4-s − 2·9-s + 2·11-s + 16-s + 4·29-s − 4·36-s + 4·44-s + 49-s − 2·64-s − 4·71-s − 2·79-s + 3·81-s − 4·99-s + 4·109-s + 8·116-s + 3·121-s + 127-s + 131-s + 137-s + 139-s − 2·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + ⋯ |
| L(s) = 1 | + 2·4-s − 2·9-s + 2·11-s + 16-s + 4·29-s − 4·36-s + 4·44-s + 49-s − 2·64-s − 4·71-s − 2·79-s + 3·81-s − 4·99-s + 4·109-s + 8·116-s + 3·121-s + 127-s + 131-s + 137-s + 139-s − 2·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.063654534\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.063654534\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| good | 2 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 13 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 17 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 47 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 83 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.04617732380829806092168981110, −6.51033319227279921106186720480, −6.37228678742432010156662628566, −6.36043914413746361929217544352, −6.30533529410754720474081293563, −5.79949499433153945011159035881, −5.79071754921210242733575081470, −5.77847644038558579992083573489, −5.19468295254110238710123648436, −4.82148606476732153881546481254, −4.72639573927937808472050199215, −4.67433901265314038003178145429, −4.14384563040902869233923008956, −4.02025632214192708359342399432, −3.88473803549106906982580107473, −3.22163759071610718594362779710, −3.01694497752613453450060376560, −2.98671574054719098737459256306, −2.89592097041120463039240858315, −2.57117304024433075536279854819, −2.05669390229901320017423026929, −1.98492320336306636557745181058, −1.65675978560010566305981127842, −1.07673924168719496661651400743, −0.933100740722115110091882544548,
0.933100740722115110091882544548, 1.07673924168719496661651400743, 1.65675978560010566305981127842, 1.98492320336306636557745181058, 2.05669390229901320017423026929, 2.57117304024433075536279854819, 2.89592097041120463039240858315, 2.98671574054719098737459256306, 3.01694497752613453450060376560, 3.22163759071610718594362779710, 3.88473803549106906982580107473, 4.02025632214192708359342399432, 4.14384563040902869233923008956, 4.67433901265314038003178145429, 4.72639573927937808472050199215, 4.82148606476732153881546481254, 5.19468295254110238710123648436, 5.77847644038558579992083573489, 5.79071754921210242733575081470, 5.79949499433153945011159035881, 6.30533529410754720474081293563, 6.36043914413746361929217544352, 6.37228678742432010156662628566, 6.51033319227279921106186720480, 7.04617732380829806092168981110
