L(s) = 1 | − 0.116·2-s + 3-s − 1.98·4-s − 1.39·5-s − 0.116·6-s − 4.81·7-s + 0.465·8-s + 9-s + 0.162·10-s + 1.77·11-s − 1.98·12-s + 13-s + 0.562·14-s − 1.39·15-s + 3.91·16-s − 0.422·17-s − 0.116·18-s + 1.32·19-s + 2.76·20-s − 4.81·21-s − 0.207·22-s + 7.34·23-s + 0.465·24-s − 3.05·25-s − 0.116·26-s + 27-s + 9.55·28-s + ⋯ |
L(s) = 1 | − 0.0826·2-s + 0.577·3-s − 0.993·4-s − 0.623·5-s − 0.0477·6-s − 1.81·7-s + 0.164·8-s + 0.333·9-s + 0.0514·10-s + 0.536·11-s − 0.573·12-s + 0.277·13-s + 0.150·14-s − 0.359·15-s + 0.979·16-s − 0.102·17-s − 0.0275·18-s + 0.305·19-s + 0.618·20-s − 1.05·21-s − 0.0443·22-s + 1.53·23-s + 0.0950·24-s − 0.611·25-s − 0.0229·26-s + 0.192·27-s + 1.80·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
good | 2 | \( 1 + 0.116T + 2T^{2} \) |
| 5 | \( 1 + 1.39T + 5T^{2} \) |
| 7 | \( 1 + 4.81T + 7T^{2} \) |
| 11 | \( 1 - 1.77T + 11T^{2} \) |
| 17 | \( 1 + 0.422T + 17T^{2} \) |
| 19 | \( 1 - 1.32T + 19T^{2} \) |
| 23 | \( 1 - 7.34T + 23T^{2} \) |
| 29 | \( 1 + 1.65T + 29T^{2} \) |
| 31 | \( 1 - 8.80T + 31T^{2} \) |
| 37 | \( 1 - 0.494T + 37T^{2} \) |
| 41 | \( 1 - 5.35T + 41T^{2} \) |
| 43 | \( 1 - 1.57T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 + 8.41T + 53T^{2} \) |
| 59 | \( 1 + 8.75T + 59T^{2} \) |
| 61 | \( 1 + 3.98T + 61T^{2} \) |
| 67 | \( 1 + 2.56T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 - 12.6T + 79T^{2} \) |
| 83 | \( 1 - 3.36T + 83T^{2} \) |
| 89 | \( 1 - 10.1T + 89T^{2} \) |
| 97 | \( 1 + 10.2T + 97T^{2} \) |
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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
−8.123268524976489854463971423373, −7.49694092991191391052207628788, −6.59568781869381478339326059669, −6.03162678928835680665014287769, −4.85209818161597032944414023818, −4.11070278964453675501427399810, −3.36983757234734097672279719528, −2.90881067077833735296499738152, −1.15867481365749911310454277658, 0,
1.15867481365749911310454277658, 2.90881067077833735296499738152, 3.36983757234734097672279719528, 4.11070278964453675501427399810, 4.85209818161597032944414023818, 6.03162678928835680665014287769, 6.59568781869381478339326059669, 7.49694092991191391052207628788, 8.123268524976489854463971423373