L(s) = 1 | + 5-s − 7-s − 0.627·11-s − 1.37·13-s − 5.37·17-s + 6.74·19-s − 6.74·23-s + 25-s − 1.37·29-s − 8·31-s − 35-s − 2·37-s + 4.74·41-s + 2.74·43-s − 10.1·47-s + 49-s + 0.744·53-s − 0.627·55-s − 8·59-s + 8.74·61-s − 1.37·65-s − 4·67-s − 8·71-s − 6·73-s + 0.627·77-s − 2.11·79-s − 13.4·83-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.377·7-s − 0.189·11-s − 0.380·13-s − 1.30·17-s + 1.54·19-s − 1.40·23-s + 0.200·25-s − 0.254·29-s − 1.43·31-s − 0.169·35-s − 0.328·37-s + 0.740·41-s + 0.418·43-s − 1.47·47-s + 0.142·49-s + 0.102·53-s − 0.0846·55-s − 1.04·59-s + 1.11·61-s − 0.170·65-s − 0.488·67-s − 0.949·71-s − 0.702·73-s + 0.0715·77-s − 0.238·79-s − 1.48·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 + 0.627T + 11T^{2} \) |
| 13 | \( 1 + 1.37T + 13T^{2} \) |
| 17 | \( 1 + 5.37T + 17T^{2} \) |
| 19 | \( 1 - 6.74T + 19T^{2} \) |
| 23 | \( 1 + 6.74T + 23T^{2} \) |
| 29 | \( 1 + 1.37T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 4.74T + 41T^{2} \) |
| 43 | \( 1 - 2.74T + 43T^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 53 | \( 1 - 0.744T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 - 8.74T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 2.11T + 79T^{2} \) |
| 83 | \( 1 + 13.4T + 83T^{2} \) |
| 89 | \( 1 + 3.25T + 89T^{2} \) |
| 97 | \( 1 - 18.8T + 97T^{2} \) |
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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
−8.639023573193590897685468693556, −7.64771707865377070173323032058, −7.06479948694465976360463360323, −6.12994451314874211855368659135, −5.49336908063962561400535550776, −4.57906171367077215620952219998, −3.61140589103804477795759636497, −2.62086274091989439536062087091, −1.65755016795417686027491303856, 0,
1.65755016795417686027491303856, 2.62086274091989439536062087091, 3.61140589103804477795759636497, 4.57906171367077215620952219998, 5.49336908063962561400535550776, 6.12994451314874211855368659135, 7.06479948694465976360463360323, 7.64771707865377070173323032058, 8.639023573193590897685468693556