| L(s) = 1 | − 2.23·2-s − 1.34·3-s + 2.98·4-s − 5-s + 2.99·6-s + 4.13·7-s − 2.19·8-s − 1.19·9-s + 2.23·10-s − 11-s − 4.00·12-s − 0.110·13-s − 9.22·14-s + 1.34·15-s − 1.07·16-s − 6.99·17-s + 2.67·18-s − 19-s − 2.98·20-s − 5.54·21-s + 2.23·22-s + 8.36·23-s + 2.94·24-s + 25-s + 0.246·26-s + 5.63·27-s + 12.3·28-s + ⋯ |
| L(s) = 1 | − 1.57·2-s − 0.775·3-s + 1.49·4-s − 0.447·5-s + 1.22·6-s + 1.56·7-s − 0.774·8-s − 0.399·9-s + 0.705·10-s − 0.301·11-s − 1.15·12-s − 0.0306·13-s − 2.46·14-s + 0.346·15-s − 0.268·16-s − 1.69·17-s + 0.630·18-s − 0.229·19-s − 0.666·20-s − 1.21·21-s + 0.475·22-s + 1.74·23-s + 0.600·24-s + 0.200·25-s + 0.0483·26-s + 1.08·27-s + 2.32·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 2.23T + 2T^{2} \) |
| 3 | \( 1 + 1.34T + 3T^{2} \) |
| 7 | \( 1 - 4.13T + 7T^{2} \) |
| 13 | \( 1 + 0.110T + 13T^{2} \) |
| 17 | \( 1 + 6.99T + 17T^{2} \) |
| 23 | \( 1 - 8.36T + 23T^{2} \) |
| 29 | \( 1 + 3.28T + 29T^{2} \) |
| 31 | \( 1 + 0.202T + 31T^{2} \) |
| 37 | \( 1 + 0.683T + 37T^{2} \) |
| 41 | \( 1 + 4.97T + 41T^{2} \) |
| 43 | \( 1 - 8.52T + 43T^{2} \) |
| 47 | \( 1 - 3.85T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 + 3.53T + 61T^{2} \) |
| 67 | \( 1 + 7.84T + 67T^{2} \) |
| 71 | \( 1 + 1.58T + 71T^{2} \) |
| 73 | \( 1 - 8.41T + 73T^{2} \) |
| 79 | \( 1 - 2.80T + 79T^{2} \) |
| 83 | \( 1 + 5.52T + 83T^{2} \) |
| 89 | \( 1 - 0.654T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−9.210639339524858782577512041664, −8.749591151119316868879809575963, −7.988489097840315838006775782461, −7.27582251087842230925541868575, −6.41548201123471472902470690455, −5.14532073431083895601257272489, −4.47882778755237386319706651076, −2.56888487282632269075833734663, −1.36558628885804903977138240660, 0,
1.36558628885804903977138240660, 2.56888487282632269075833734663, 4.47882778755237386319706651076, 5.14532073431083895601257272489, 6.41548201123471472902470690455, 7.27582251087842230925541868575, 7.988489097840315838006775782461, 8.749591151119316868879809575963, 9.210639339524858782577512041664
