| L(s) = 1 | − 5-s − 2·7-s + 4·11-s + 13-s − 4·19-s + 23-s + 25-s + 7·29-s − 7·31-s + 2·35-s − 4·37-s − 3·41-s + 6·43-s + 13·47-s − 3·49-s − 10·53-s − 4·55-s + 8·59-s − 65-s + 8·67-s − 13·71-s + 11·73-s − 8·77-s + 4·79-s + 4·83-s + 6·89-s − 2·91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s + 1.20·11-s + 0.277·13-s − 0.917·19-s + 0.208·23-s + 1/5·25-s + 1.29·29-s − 1.25·31-s + 0.338·35-s − 0.657·37-s − 0.468·41-s + 0.914·43-s + 1.89·47-s − 3/7·49-s − 1.37·53-s − 0.539·55-s + 1.04·59-s − 0.124·65-s + 0.977·67-s − 1.54·71-s + 1.28·73-s − 0.911·77-s + 0.450·79-s + 0.439·83-s + 0.635·89-s − 0.209·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.530741149\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.530741149\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.582977332348146562710091043065, −7.64570064973687093757758439724, −6.79272605407006166458827493072, −6.42501841167079437554065819928, −5.53392590775861732995702419988, −4.46792999426338875485798833452, −3.82744709866793850051848950489, −3.10614689114143817615345596705, −1.94178657204270397883871050919, −0.70427278926568134058720289386,
0.70427278926568134058720289386, 1.94178657204270397883871050919, 3.10614689114143817615345596705, 3.82744709866793850051848950489, 4.46792999426338875485798833452, 5.53392590775861732995702419988, 6.42501841167079437554065819928, 6.79272605407006166458827493072, 7.64570064973687093757758439724, 8.582977332348146562710091043065
