| L(s) = 1 | + 4.60·7-s − 11-s − 1.60·13-s − 4.60·17-s + 6.60·19-s + 1.60·23-s − 6.60·29-s − 2·31-s + 5.60·37-s + 6·41-s + 1.39·43-s + 12.8·47-s + 14.2·49-s − 7.81·53-s + 12.2·59-s − 2.39·61-s − 0.605·67-s − 11.6·71-s + 6·73-s − 4.60·77-s + 8.60·79-s + 9.21·83-s + 8.60·89-s − 7.39·91-s − 4.21·97-s + 13.8·101-s + 15.2·103-s + ⋯ |
| L(s) = 1 | + 1.74·7-s − 0.301·11-s − 0.445·13-s − 1.11·17-s + 1.51·19-s + 0.334·23-s − 1.22·29-s − 0.359·31-s + 0.921·37-s + 0.937·41-s + 0.212·43-s + 1.86·47-s + 2.03·49-s − 1.07·53-s + 1.58·59-s − 0.306·61-s − 0.0739·67-s − 1.37·71-s + 0.702·73-s − 0.524·77-s + 0.968·79-s + 1.01·83-s + 0.912·89-s − 0.775·91-s − 0.427·97-s + 1.37·101-s + 1.49·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.430381289\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.430381289\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 4.60T + 7T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 + 1.60T + 13T^{2} \) |
| 17 | \( 1 + 4.60T + 17T^{2} \) |
| 19 | \( 1 - 6.60T + 19T^{2} \) |
| 23 | \( 1 - 1.60T + 23T^{2} \) |
| 29 | \( 1 + 6.60T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 5.60T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 1.39T + 43T^{2} \) |
| 47 | \( 1 - 12.8T + 47T^{2} \) |
| 53 | \( 1 + 7.81T + 53T^{2} \) |
| 59 | \( 1 - 12.2T + 59T^{2} \) |
| 61 | \( 1 + 2.39T + 61T^{2} \) |
| 67 | \( 1 + 0.605T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 8.60T + 79T^{2} \) |
| 83 | \( 1 - 9.21T + 83T^{2} \) |
| 89 | \( 1 - 8.60T + 89T^{2} \) |
| 97 | \( 1 + 4.21T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.85042622932739229864515388486, −7.71657516937312479395288617196, −6.94584567902336210747227924607, −5.81010550152880775139144115558, −5.22185841546652919044362756000, −4.60773007475626505369494800816, −3.85077292500479444592129151404, −2.62999292783074479636298784041, −1.91799235355693400511741515918, −0.866375467572822875373947351124,
0.866375467572822875373947351124, 1.91799235355693400511741515918, 2.62999292783074479636298784041, 3.85077292500479444592129151404, 4.60773007475626505369494800816, 5.22185841546652919044362756000, 5.81010550152880775139144115558, 6.94584567902336210747227924607, 7.71657516937312479395288617196, 7.85042622932739229864515388486
