L(s) = 1 | − 2·5-s + 7-s + 11-s + 6·13-s − 6·17-s − 8·23-s − 25-s + 2·29-s − 4·31-s − 2·35-s − 2·37-s + 2·41-s + 12·43-s + 12·47-s + 49-s − 6·53-s − 2·55-s + 4·59-s + 6·61-s − 12·65-s − 4·67-s − 2·73-s + 77-s − 8·79-s + 12·85-s + 6·89-s + 6·91-s + ⋯ |
L(s) = 1 | − 0.894·5-s + 0.377·7-s + 0.301·11-s + 1.66·13-s − 1.45·17-s − 1.66·23-s − 1/5·25-s + 0.371·29-s − 0.718·31-s − 0.338·35-s − 0.328·37-s + 0.312·41-s + 1.82·43-s + 1.75·47-s + 1/7·49-s − 0.824·53-s − 0.269·55-s + 0.520·59-s + 0.768·61-s − 1.48·65-s − 0.488·67-s − 0.234·73-s + 0.113·77-s − 0.900·79-s + 1.30·85-s + 0.635·89-s + 0.628·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11088 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.579031123\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.579031123\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.11502086550629, −15.86118291421270, −15.66759451252851, −14.80605348949853, −14.14709744198393, −13.71968385217766, −13.08140369384830, −12.39982310840207, −11.77901194824096, −11.29339528696083, −10.82765739405334, −10.26629163859589, −9.204624442018878, −8.822295188148051, −8.206256907762717, −7.648696668615248, −6.963224711565187, −6.123531471818431, −5.770411328766132, −4.603674733702279, −3.990874585829314, −3.722352793025873, −2.496407378647474, −1.679598128453642, −0.5879239591642018,
0.5879239591642018, 1.679598128453642, 2.496407378647474, 3.722352793025873, 3.990874585829314, 4.603674733702279, 5.770411328766132, 6.123531471818431, 6.963224711565187, 7.648696668615248, 8.206256907762717, 8.822295188148051, 9.204624442018878, 10.26629163859589, 10.82765739405334, 11.29339528696083, 11.77901194824096, 12.39982310840207, 13.08140369384830, 13.71968385217766, 14.14709744198393, 14.80605348949853, 15.66759451252851, 15.86118291421270, 16.11502086550629