| L(s) = 1 | − 5-s + 2·13-s − 6·17-s + 4·19-s + 25-s + 6·29-s − 8·31-s + 6·37-s + 10·41-s + 4·43-s + 8·47-s − 7·49-s + 10·53-s + 12·59-s − 6·61-s − 2·65-s − 4·67-s + 14·73-s + 4·83-s + 6·85-s + 6·89-s − 4·95-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.554·13-s − 1.45·17-s + 0.917·19-s + 1/5·25-s + 1.11·29-s − 1.43·31-s + 0.986·37-s + 1.56·41-s + 0.609·43-s + 1.16·47-s − 49-s + 1.37·53-s + 1.56·59-s − 0.768·61-s − 0.248·65-s − 0.488·67-s + 1.63·73-s + 0.439·83-s + 0.650·85-s + 0.635·89-s − 0.410·95-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.058649246\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.058649246\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−14.75627131379015, −14.15803668035169, −13.56310964509067, −13.25100466706845, −12.54395116457882, −12.16595640341302, −11.47139759383822, −10.96709212514682, −10.79530770980866, −9.916211201889725, −9.316368742229620, −8.927516534143440, −8.338945274089098, −7.731824158563253, −7.215723127978143, −6.655282687732455, −6.042765446101765, −5.437709437526718, −4.768665210278256, −4.065945358865158, −3.748277698998339, −2.748564474274023, −2.326004332379765, −1.267417065142521, −0.5593691695150149,
0.5593691695150149, 1.267417065142521, 2.326004332379765, 2.748564474274023, 3.748277698998339, 4.065945358865158, 4.768665210278256, 5.437709437526718, 6.042765446101765, 6.655282687732455, 7.215723127978143, 7.731824158563253, 8.338945274089098, 8.927516534143440, 9.316368742229620, 9.916211201889725, 10.79530770980866, 10.96709212514682, 11.47139759383822, 12.16595640341302, 12.54395116457882, 13.25100466706845, 13.56310964509067, 14.15803668035169, 14.75627131379015
