| L(s) = 1 | − 2·3-s + 2·5-s + 7-s + 9-s − 2·13-s − 4·15-s − 2·17-s − 19-s − 2·21-s − 4·23-s − 25-s + 4·27-s − 8·29-s + 2·35-s − 10·37-s + 4·39-s − 2·41-s − 4·43-s + 2·45-s − 47-s + 49-s + 4·51-s − 6·53-s + 2·57-s − 10·59-s + 14·61-s + 63-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s − 0.554·13-s − 1.03·15-s − 0.485·17-s − 0.229·19-s − 0.436·21-s − 0.834·23-s − 1/5·25-s + 0.769·27-s − 1.48·29-s + 0.338·35-s − 1.64·37-s + 0.640·39-s − 0.312·41-s − 0.609·43-s + 0.298·45-s − 0.145·47-s + 1/7·49-s + 0.560·51-s − 0.824·53-s + 0.264·57-s − 1.30·59-s + 1.79·61-s + 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1075974915\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1075974915\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 47 | \( 1 + T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−13.66422753329309, −13.28439073836982, −12.71951696089543, −12.17193564025107, −11.82520841476332, −11.27681799519984, −10.88608131328861, −10.30757593161126, −9.970333837432331, −9.411283132961243, −8.815179182932072, −8.345084041137652, −7.593425764067315, −7.139644392605719, −6.448523329096879, −6.158768729213164, −5.420256921559849, −5.319096636740958, −4.617539600455230, −4.025201659060028, −3.265815275989085, −2.442210255531693, −1.825805328207019, −1.385146123144242, −0.1065124273002247,
0.1065124273002247, 1.385146123144242, 1.825805328207019, 2.442210255531693, 3.265815275989085, 4.025201659060028, 4.617539600455230, 5.319096636740958, 5.420256921559849, 6.158768729213164, 6.448523329096879, 7.139644392605719, 7.593425764067315, 8.345084041137652, 8.815179182932072, 9.411283132961243, 9.970333837432331, 10.30757593161126, 10.88608131328861, 11.27681799519984, 11.82520841476332, 12.17193564025107, 12.71951696089543, 13.28439073836982, 13.66422753329309
