| L(s) = 1 | − 2·4-s − 8·7-s + 8·13-s + 4·16-s + 2·19-s − 8·25-s + 16·28-s + 14·31-s − 16·37-s + 2·43-s + 34·49-s − 16·52-s − 10·61-s − 16·64-s − 4·67-s + 2·73-s − 4·76-s + 8·79-s − 64·91-s + 2·97-s + 16·100-s + 8·103-s + 2·109-s − 32·112-s + 4·121-s − 28·124-s + 127-s + ⋯ |
| L(s) = 1 | − 4-s − 3.02·7-s + 2.21·13-s + 16-s + 0.458·19-s − 8/5·25-s + 3.02·28-s + 2.51·31-s − 2.63·37-s + 0.304·43-s + 34/7·49-s − 2.21·52-s − 1.28·61-s − 2·64-s − 0.488·67-s + 0.234·73-s − 0.458·76-s + 0.900·79-s − 6.70·91-s + 0.203·97-s + 8/5·100-s + 0.788·103-s + 0.191·109-s − 3.02·112-s + 4/11·121-s − 2.51·124-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3770153081\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3770153081\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| good | 2 | $C_2$$\times$$C_2^2$ | \( ( 1 + p T^{2} )^{2}( 1 - p T^{2} + p^{2} T^{4} ) \) |
| 5 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 28 T^{2} + 495 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^3$ | \( 1 - 4 T^{2} - 825 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^3$ | \( 1 - 28 T^{2} - 897 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 88 T^{2} + 5535 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 100 T^{2} + 7191 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^3$ | \( 1 - 22 T^{2} - 2997 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - T - 72 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 50 T^{2} - 4389 T^{4} + 50 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^3$ | \( 1 - 172 T^{2} + 21663 T^{4} - 172 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - T - 96 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.61737442359889633488216984919, −7.59083223823767027602992311861, −7.25522463134601351351674120180, −6.89868850903446079616109056168, −6.83229929106983637486150631690, −6.45799528099584254834729000613, −6.12069058693647968868658808981, −6.01578797097937647725653332318, −5.97478887496319383165699387717, −5.90973489970477205169761434699, −5.22931488387540024484848881506, −5.07943859515922622071576289099, −4.83424975606445727526298078163, −4.26466515054136602391586822191, −4.09869361629574243671640701375, −3.99644598458172007022110495082, −3.45976135762304205244811735791, −3.26310415655367670444175769781, −3.18441460120769630009207096785, −3.13396817996704651819986081876, −2.29510795371408496686146749530, −2.10385711133703611604695352493, −1.24354019810695354576710629675, −1.07963498738593615824902812933, −0.23121431181325405628978765945,
0.23121431181325405628978765945, 1.07963498738593615824902812933, 1.24354019810695354576710629675, 2.10385711133703611604695352493, 2.29510795371408496686146749530, 3.13396817996704651819986081876, 3.18441460120769630009207096785, 3.26310415655367670444175769781, 3.45976135762304205244811735791, 3.99644598458172007022110495082, 4.09869361629574243671640701375, 4.26466515054136602391586822191, 4.83424975606445727526298078163, 5.07943859515922622071576289099, 5.22931488387540024484848881506, 5.90973489970477205169761434699, 5.97478887496319383165699387717, 6.01578797097937647725653332318, 6.12069058693647968868658808981, 6.45799528099584254834729000613, 6.83229929106983637486150631690, 6.89868850903446079616109056168, 7.25522463134601351351674120180, 7.59083223823767027602992311861, 7.61737442359889633488216984919
