| L(s) = 1 | + 2·7-s − 25-s − 6·31-s + 49-s + 2·79-s − 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s − 2·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s − 12·217-s + 223-s + ⋯ |
| L(s) = 1 | + 2·7-s − 25-s − 6·31-s + 49-s + 2·79-s − 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s − 2·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s − 12·217-s + 223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.207205376\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.207205376\) |
| \(L(1)\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| good | 5 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 53 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 61 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
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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
−6.77017211782634985405107076379, −6.60319234581935112019081635738, −6.31254023219666110765394716893, −5.84325780958923743493269338971, −5.72537615874244332257885355216, −5.66040613165431034923401088686, −5.49801486711229229540776655991, −5.30257849445462530020878861480, −5.00001713093185596057047142768, −4.80731484587997741225077686461, −4.62312554575023135357416503864, −4.37773044823573934479972895224, −4.15623370320562464879465113375, −3.76390333179763627730854732260, −3.63838132946721242100265009044, −3.45555184705016807768510337057, −3.40658408062262507483547913659, −2.85259609683907292955774033930, −2.50744296080747164808441305408, −2.10298852297526176634389596468, −2.04175386152273353527504027886, −1.75349527851813481850706749980, −1.59517689284999459059657863235, −1.28797238410692487110327069689, −0.52528477180275316985364138298,
0.52528477180275316985364138298, 1.28797238410692487110327069689, 1.59517689284999459059657863235, 1.75349527851813481850706749980, 2.04175386152273353527504027886, 2.10298852297526176634389596468, 2.50744296080747164808441305408, 2.85259609683907292955774033930, 3.40658408062262507483547913659, 3.45555184705016807768510337057, 3.63838132946721242100265009044, 3.76390333179763627730854732260, 4.15623370320562464879465113375, 4.37773044823573934479972895224, 4.62312554575023135357416503864, 4.80731484587997741225077686461, 5.00001713093185596057047142768, 5.30257849445462530020878861480, 5.49801486711229229540776655991, 5.66040613165431034923401088686, 5.72537615874244332257885355216, 5.84325780958923743493269338971, 6.31254023219666110765394716893, 6.60319234581935112019081635738, 6.77017211782634985405107076379
