L(s) = 1 | + 2-s − 2.34·3-s + 4-s − 5-s − 2.34·6-s − 0.317·7-s + 8-s + 2.51·9-s − 10-s + 0.699·11-s − 2.34·12-s − 1.60·13-s − 0.317·14-s + 2.34·15-s + 16-s + 0.615·17-s + 2.51·18-s + 1.67·19-s − 20-s + 0.745·21-s + 0.699·22-s − 4.91·23-s − 2.34·24-s + 25-s − 1.60·26-s + 1.12·27-s − 0.317·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.35·3-s + 0.5·4-s − 0.447·5-s − 0.959·6-s − 0.119·7-s + 0.353·8-s + 0.839·9-s − 0.316·10-s + 0.210·11-s − 0.678·12-s − 0.444·13-s − 0.0847·14-s + 0.606·15-s + 0.250·16-s + 0.149·17-s + 0.593·18-s + 0.385·19-s − 0.223·20-s + 0.162·21-s + 0.149·22-s − 1.02·23-s − 0.479·24-s + 0.200·25-s − 0.314·26-s + 0.217·27-s − 0.0599·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
good | 3 | \( 1 + 2.34T + 3T^{2} \) |
| 7 | \( 1 + 0.317T + 7T^{2} \) |
| 11 | \( 1 - 0.699T + 11T^{2} \) |
| 13 | \( 1 + 1.60T + 13T^{2} \) |
| 17 | \( 1 - 0.615T + 17T^{2} \) |
| 19 | \( 1 - 1.67T + 19T^{2} \) |
| 23 | \( 1 + 4.91T + 23T^{2} \) |
| 29 | \( 1 + 1.51T + 29T^{2} \) |
| 31 | \( 1 - 9.27T + 31T^{2} \) |
| 37 | \( 1 + 2.30T + 37T^{2} \) |
| 41 | \( 1 - 2.64T + 41T^{2} \) |
| 43 | \( 1 + 2.37T + 43T^{2} \) |
| 47 | \( 1 + 8.32T + 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 + 3.35T + 59T^{2} \) |
| 61 | \( 1 - 0.448T + 61T^{2} \) |
| 67 | \( 1 - 0.654T + 67T^{2} \) |
| 71 | \( 1 - 5.61T + 71T^{2} \) |
| 73 | \( 1 + 8.44T + 73T^{2} \) |
| 79 | \( 1 + 8.36T + 79T^{2} \) |
| 83 | \( 1 + 12.6T + 83T^{2} \) |
| 89 | \( 1 + 0.0408T + 89T^{2} \) |
| 97 | \( 1 + 10.6T + 97T^{2} \) |
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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
−7.86503721033244700839666001442, −7.09130790000801712112181988080, −6.40795207935627179494024462454, −5.85087709706781833631346385277, −5.05423945047027675404738889358, −4.49496921617349174391190817533, −3.62708605499536321197424263781, −2.61474193370765391059020731105, −1.27708684416439217718869980421, 0,
1.27708684416439217718869980421, 2.61474193370765391059020731105, 3.62708605499536321197424263781, 4.49496921617349174391190817533, 5.05423945047027675404738889358, 5.85087709706781833631346385277, 6.40795207935627179494024462454, 7.09130790000801712112181988080, 7.86503721033244700839666001442