| 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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 4.21T + 97T^{2} \) |
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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
−7.58742126986816086080078664979, −7.22709803552073335339656186577, −6.29756223271206471315425837462, −5.73301005927962657422782830482, −5.09301308218163237708290534026, −3.66358235882379809842775611968, −3.48491499882723075782699420354, −2.55265678300263577218596538712, −1.22566651541307794072633206217, 0,
1.22566651541307794072633206217, 2.55265678300263577218596538712, 3.48491499882723075782699420354, 3.66358235882379809842775611968, 5.09301308218163237708290534026, 5.73301005927962657422782830482, 6.29756223271206471315425837462, 7.22709803552073335339656186577, 7.58742126986816086080078664979
