| L(s) = 1 | + 4·5-s + 7-s + 2·13-s − 2·19-s + 4·23-s + 11·25-s + 6·29-s + 10·31-s + 4·35-s + 2·37-s + 8·41-s + 2·47-s + 49-s − 6·53-s + 8·59-s + 10·61-s + 8·65-s + 12·67-s + 8·71-s − 8·73-s − 8·79-s + 10·83-s + 6·89-s + 2·91-s − 8·95-s + 10·97-s + 101-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 0.377·7-s + 0.554·13-s − 0.458·19-s + 0.834·23-s + 11/5·25-s + 1.11·29-s + 1.79·31-s + 0.676·35-s + 0.328·37-s + 1.24·41-s + 0.291·47-s + 1/7·49-s − 0.824·53-s + 1.04·59-s + 1.28·61-s + 0.992·65-s + 1.46·67-s + 0.949·71-s − 0.936·73-s − 0.900·79-s + 1.09·83-s + 0.635·89-s + 0.209·91-s − 0.820·95-s + 1.01·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.388141300\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.388141300\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.19693133024291, −13.88921475659942, −13.26183711975618, −12.92102995079045, −12.48011862791190, −11.64545687089059, −11.24373752035534, −10.54847036498941, −10.25477757323574, −9.709408584354091, −9.230698006645692, −8.605983295367883, −8.307042646004361, −7.501083993005268, −6.711735698664201, −6.372191272979413, −5.950531656112192, −5.218375594914558, −4.849444628610963, −4.169951181456715, −3.287861670507482, −2.427496530708235, −2.324193196276720, −1.199684133856859, −0.9266703042461379,
0.9266703042461379, 1.199684133856859, 2.324193196276720, 2.427496530708235, 3.287861670507482, 4.169951181456715, 4.849444628610963, 5.218375594914558, 5.950531656112192, 6.372191272979413, 6.711735698664201, 7.501083993005268, 8.307042646004361, 8.605983295367883, 9.230698006645692, 9.709408584354091, 10.25477757323574, 10.54847036498941, 11.24373752035534, 11.64545687089059, 12.48011862791190, 12.92102995079045, 13.26183711975618, 13.88921475659942, 14.19693133024291
