L(s) = 1 | + 2.68·2-s + 3-s + 5.20·4-s − 1.64·5-s + 2.68·6-s + 7-s + 8.60·8-s + 9-s − 4.42·10-s − 1.55·11-s + 5.20·12-s + 1.36·13-s + 2.68·14-s − 1.64·15-s + 12.6·16-s + 3.98·17-s + 2.68·18-s + 4.09·19-s − 8.57·20-s + 21-s − 4.17·22-s − 1.17·23-s + 8.60·24-s − 2.28·25-s + 3.65·26-s + 27-s + 5.20·28-s + ⋯ |
L(s) = 1 | + 1.89·2-s + 0.577·3-s + 2.60·4-s − 0.736·5-s + 1.09·6-s + 0.377·7-s + 3.04·8-s + 0.333·9-s − 1.39·10-s − 0.469·11-s + 1.50·12-s + 0.378·13-s + 0.717·14-s − 0.425·15-s + 3.17·16-s + 0.965·17-s + 0.632·18-s + 0.938·19-s − 1.91·20-s + 0.218·21-s − 0.890·22-s − 0.245·23-s + 1.75·24-s − 0.457·25-s + 0.717·26-s + 0.192·27-s + 0.983·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(8.027555863\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.027555863\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 191 | \( 1 - T \) |
good | 2 | \( 1 - 2.68T + 2T^{2} \) |
| 5 | \( 1 + 1.64T + 5T^{2} \) |
| 11 | \( 1 + 1.55T + 11T^{2} \) |
| 13 | \( 1 - 1.36T + 13T^{2} \) |
| 17 | \( 1 - 3.98T + 17T^{2} \) |
| 19 | \( 1 - 4.09T + 19T^{2} \) |
| 23 | \( 1 + 1.17T + 23T^{2} \) |
| 29 | \( 1 + 1.85T + 29T^{2} \) |
| 31 | \( 1 - 6.08T + 31T^{2} \) |
| 37 | \( 1 + 10.4T + 37T^{2} \) |
| 41 | \( 1 + 1.99T + 41T^{2} \) |
| 43 | \( 1 - 3.05T + 43T^{2} \) |
| 47 | \( 1 - 7.18T + 47T^{2} \) |
| 53 | \( 1 - 12.6T + 53T^{2} \) |
| 59 | \( 1 + 5.41T + 59T^{2} \) |
| 61 | \( 1 + 12.9T + 61T^{2} \) |
| 67 | \( 1 + 1.92T + 67T^{2} \) |
| 71 | \( 1 + 1.64T + 71T^{2} \) |
| 73 | \( 1 + 6.10T + 73T^{2} \) |
| 79 | \( 1 + 0.436T + 79T^{2} \) |
| 83 | \( 1 - 11.6T + 83T^{2} \) |
| 89 | \( 1 - 6.60T + 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
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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
−7.980730158002649454444493828420, −7.61608800656674429947715781679, −6.95414175719927916018046653347, −5.93685962463802413018595657879, −5.34013072137048538840187481920, −4.57258238294157857313316517085, −3.79521231858957030677497860693, −3.28921034128379239376945520627, −2.45014924915856830625376957278, −1.36352369775640794316267247281,
1.36352369775640794316267247281, 2.45014924915856830625376957278, 3.28921034128379239376945520627, 3.79521231858957030677497860693, 4.57258238294157857313316517085, 5.34013072137048538840187481920, 5.93685962463802413018595657879, 6.95414175719927916018046653347, 7.61608800656674429947715781679, 7.980730158002649454444493828420