| L(s) = 1 | − 2·3-s − 2·7-s + 9-s − 2·11-s − 4·19-s + 4·21-s + 2·27-s + 4·33-s + 4·37-s − 2·41-s + 2·47-s + 2·49-s − 4·53-s + 8·57-s − 2·61-s − 2·63-s + 4·77-s − 2·79-s − 4·81-s + 2·83-s − 4·89-s − 2·99-s + 4·107-s − 4·109-s − 8·111-s + 2·113-s + 2·121-s + ⋯ |
| L(s) = 1 | − 2·3-s − 2·7-s + 9-s − 2·11-s − 4·19-s + 4·21-s + 2·27-s + 4·33-s + 4·37-s − 2·41-s + 2·47-s + 2·49-s − 4·53-s + 8·57-s − 2·61-s − 2·63-s + 4·77-s − 2·79-s − 4·81-s + 2·83-s − 4·89-s − 2·99-s + 4·107-s − 4·109-s − 8·111-s + 2·113-s + 2·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.002320931235\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.002320931235\) |
| \(L(1)\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| good | 5 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 7 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 17 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 29 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 31 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 41 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 47 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 53 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 59 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 67 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 83 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 97 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
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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
−7.30971694400707788089361549531, −7.10053378785479131063778362179, −6.81571180585292997269087912433, −6.44137643161137144815183832472, −6.41291468871899552777797448079, −6.34053909065931783032001460845, −6.11669808259269780533514737780, −6.00345416943449671778289439642, −5.56907634088591951604589899920, −5.53972520148029502310579387445, −5.31304530772700694985076424875, −4.66220801929755282276057451662, −4.66044480531168502521656369356, −4.61989787995641170815724929487, −4.18188364311404514509478414013, −4.11387808581167344210765972342, −3.53894067762455677607740541711, −3.15925200030540754546133380370, −2.96153207448637234642334022329, −2.72446859138517158782270158406, −2.50169813500570620633764367576, −2.19091270186842215233725906227, −1.69037572181449195471345532516, −1.02356671019582298785442091026, −0.05052041601312060385186691248,
0.05052041601312060385186691248, 1.02356671019582298785442091026, 1.69037572181449195471345532516, 2.19091270186842215233725906227, 2.50169813500570620633764367576, 2.72446859138517158782270158406, 2.96153207448637234642334022329, 3.15925200030540754546133380370, 3.53894067762455677607740541711, 4.11387808581167344210765972342, 4.18188364311404514509478414013, 4.61989787995641170815724929487, 4.66044480531168502521656369356, 4.66220801929755282276057451662, 5.31304530772700694985076424875, 5.53972520148029502310579387445, 5.56907634088591951604589899920, 6.00345416943449671778289439642, 6.11669808259269780533514737780, 6.34053909065931783032001460845, 6.41291468871899552777797448079, 6.44137643161137144815183832472, 6.81571180585292997269087912433, 7.10053378785479131063778362179, 7.30971694400707788089361549531
