| L(s) = 1 | − 2·5-s + 2·13-s + 2·17-s + 4·19-s − 25-s + 6·29-s − 6·37-s − 6·41-s − 8·43-s + 8·47-s + 6·53-s + 12·59-s + 10·61-s − 4·65-s − 16·67-s + 8·71-s + 6·73-s + 8·79-s + 12·83-s − 4·85-s − 14·89-s − 8·95-s + 6·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.554·13-s + 0.485·17-s + 0.917·19-s − 1/5·25-s + 1.11·29-s − 0.986·37-s − 0.937·41-s − 1.21·43-s + 1.16·47-s + 0.824·53-s + 1.56·59-s + 1.28·61-s − 0.496·65-s − 1.95·67-s + 0.949·71-s + 0.702·73-s + 0.900·79-s + 1.31·83-s − 0.433·85-s − 1.48·89-s − 0.820·95-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.818297448\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.818297448\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−15.33842239103637, −14.78122576061817, −14.08081326167097, −13.63358235763024, −13.21744031768102, −12.24189648710202, −12.08214812040246, −11.56522143895670, −10.99629610890960, −10.26974527322745, −9.961811809771931, −9.141169585240682, −8.508560224725059, −8.153161699715394, −7.503666248459526, −6.928267997600280, −6.415958514780344, −5.469433625855267, −5.160092552486627, −4.249505471972230, −3.669012007622061, −3.215485107115924, −2.318794616806760, −1.340859466238554, −0.5649053629205346,
0.5649053629205346, 1.340859466238554, 2.318794616806760, 3.215485107115924, 3.669012007622061, 4.249505471972230, 5.160092552486627, 5.469433625855267, 6.415958514780344, 6.928267997600280, 7.503666248459526, 8.153161699715394, 8.508560224725059, 9.141169585240682, 9.961811809771931, 10.26974527322745, 10.99629610890960, 11.56522143895670, 12.08214812040246, 12.24189648710202, 13.21744031768102, 13.63358235763024, 14.08081326167097, 14.78122576061817, 15.33842239103637
