| L(s) = 1 | − 2·5-s + 4·11-s + 2·13-s − 4·17-s − 8·19-s − 8·23-s + 5·25-s + 6·29-s − 8·31-s + 12·37-s − 6·41-s − 4·43-s + 7·49-s + 4·53-s − 8·55-s + 4·59-s + 2·61-s − 4·65-s + 4·67-s − 16·71-s + 20·73-s + 8·79-s − 4·83-s + 8·85-s + 12·89-s + 16·95-s − 2·97-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.20·11-s + 0.554·13-s − 0.970·17-s − 1.83·19-s − 1.66·23-s + 25-s + 1.11·29-s − 1.43·31-s + 1.97·37-s − 0.937·41-s − 0.609·43-s + 49-s + 0.549·53-s − 1.07·55-s + 0.520·59-s + 0.256·61-s − 0.496·65-s + 0.488·67-s − 1.89·71-s + 2.34·73-s + 0.900·79-s − 0.439·83-s + 0.867·85-s + 1.27·89-s + 1.64·95-s − 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 419904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 419904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.263001892\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.263001892\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 4 T - 67 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−10.79136450462894180523473862426, −10.39364945474135883701952701496, −10.03990724943601265756739972569, −9.228785968814071162613509608709, −9.091511287807946275349639651779, −8.572771034830756329105478232988, −8.118841486681787115399695592518, −7.958739800770420305055354198236, −7.13723303190089553070670520054, −6.58556329768194161716179141707, −6.56281557415033051199783292596, −5.94282249146933867362057864076, −5.32243520848985869924111779925, −4.49268408687844431066816523150, −4.14448695326889604007949906289, −3.95848441804194632447127503776, −3.22565559483863579379392605700, −2.32944008006588739182324639947, −1.78851140451292107198070039481, −0.61352758754528296354893035184,
0.61352758754528296354893035184, 1.78851140451292107198070039481, 2.32944008006588739182324639947, 3.22565559483863579379392605700, 3.95848441804194632447127503776, 4.14448695326889604007949906289, 4.49268408687844431066816523150, 5.32243520848985869924111779925, 5.94282249146933867362057864076, 6.56281557415033051199783292596, 6.58556329768194161716179141707, 7.13723303190089553070670520054, 7.958739800770420305055354198236, 8.118841486681787115399695592518, 8.572771034830756329105478232988, 9.091511287807946275349639651779, 9.228785968814071162613509608709, 10.03990724943601265756739972569, 10.39364945474135883701952701496, 10.79136450462894180523473862426
