L(s) = 1 | − 2-s − 3.17·3-s + 4-s + 4.15·5-s + 3.17·6-s + 1.45·7-s − 8-s + 7.05·9-s − 4.15·10-s − 3.03·11-s − 3.17·12-s + 3.91·13-s − 1.45·14-s − 13.1·15-s + 16-s − 0.0576·17-s − 7.05·18-s − 19-s + 4.15·20-s − 4.61·21-s + 3.03·22-s + 0.939·23-s + 3.17·24-s + 12.2·25-s − 3.91·26-s − 12.8·27-s + 1.45·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.83·3-s + 0.5·4-s + 1.85·5-s + 1.29·6-s + 0.549·7-s − 0.353·8-s + 2.35·9-s − 1.31·10-s − 0.915·11-s − 0.915·12-s + 1.08·13-s − 0.388·14-s − 3.40·15-s + 0.250·16-s − 0.0139·17-s − 1.66·18-s − 0.229·19-s + 0.928·20-s − 1.00·21-s + 0.647·22-s + 0.195·23-s + 0.647·24-s + 2.45·25-s − 0.768·26-s − 2.47·27-s + 0.274·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
good | 3 | \( 1 + 3.17T + 3T^{2} \) |
| 5 | \( 1 - 4.15T + 5T^{2} \) |
| 7 | \( 1 - 1.45T + 7T^{2} \) |
| 11 | \( 1 + 3.03T + 11T^{2} \) |
| 13 | \( 1 - 3.91T + 13T^{2} \) |
| 17 | \( 1 + 0.0576T + 17T^{2} \) |
| 23 | \( 1 - 0.939T + 23T^{2} \) |
| 29 | \( 1 + 9.74T + 29T^{2} \) |
| 31 | \( 1 - 3.98T + 31T^{2} \) |
| 37 | \( 1 + 10.1T + 37T^{2} \) |
| 41 | \( 1 - 6.40T + 41T^{2} \) |
| 43 | \( 1 + 0.156T + 43T^{2} \) |
| 47 | \( 1 + 3.44T + 47T^{2} \) |
| 53 | \( 1 - 8.72T + 53T^{2} \) |
| 59 | \( 1 + 6.77T + 59T^{2} \) |
| 61 | \( 1 - 0.287T + 61T^{2} \) |
| 67 | \( 1 + 0.311T + 67T^{2} \) |
| 71 | \( 1 + 4.31T + 71T^{2} \) |
| 73 | \( 1 + 12.8T + 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 + 3.11T + 83T^{2} \) |
| 89 | \( 1 + 9.56T + 89T^{2} \) |
| 97 | \( 1 + 18.1T + 97T^{2} \) |
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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
−7.21374876413022442649173587804, −6.69468181329075984050384325833, −5.94162338882312357235358614981, −5.59492126551759083668911195746, −5.15139240014627772693820006618, −4.17992800226292469045082397053, −2.76133667342133814621749843580, −1.65515114123767730542187621573, −1.34669111682136299649716726448, 0,
1.34669111682136299649716726448, 1.65515114123767730542187621573, 2.76133667342133814621749843580, 4.17992800226292469045082397053, 5.15139240014627772693820006618, 5.59492126551759083668911195746, 5.94162338882312357235358614981, 6.69468181329075984050384325833, 7.21374876413022442649173587804