| L(s) = 1 | − 1.85·2-s − 3·3-s − 4.55·4-s − 4.35·5-s + 5.56·6-s + 17.6·7-s + 23.3·8-s + 9·9-s + 8.08·10-s + 1.41·11-s + 13.6·12-s + 18.8·13-s − 32.6·14-s + 13.0·15-s − 6.81·16-s − 10.5·17-s − 16.7·18-s − 42.7·19-s + 19.8·20-s − 52.8·21-s − 2.62·22-s − 171.·23-s − 69.9·24-s − 106.·25-s − 35.0·26-s − 27·27-s − 80.2·28-s + ⋯ |
| L(s) = 1 | − 0.656·2-s − 0.577·3-s − 0.569·4-s − 0.389·5-s + 0.378·6-s + 0.950·7-s + 1.02·8-s + 0.333·9-s + 0.255·10-s + 0.0388·11-s + 0.328·12-s + 0.402·13-s − 0.624·14-s + 0.225·15-s − 0.106·16-s − 0.150·17-s − 0.218·18-s − 0.515·19-s + 0.221·20-s − 0.549·21-s − 0.0254·22-s − 1.55·23-s − 0.594·24-s − 0.848·25-s − 0.264·26-s − 0.192·27-s − 0.541·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 67 | \( 1 - 67T \) |
| good | 2 | \( 1 + 1.85T + 8T^{2} \) |
| 5 | \( 1 + 4.35T + 125T^{2} \) |
| 7 | \( 1 - 17.6T + 343T^{2} \) |
| 11 | \( 1 - 1.41T + 1.33e3T^{2} \) |
| 13 | \( 1 - 18.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 10.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 42.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 171.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 33.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 169.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 67.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 253.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 21.7T + 7.95e4T^{2} \) |
| 47 | \( 1 + 280.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 455.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 589.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 142.T + 2.26e5T^{2} \) |
| 71 | \( 1 - 140.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 106.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.12e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 173.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 754.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.03e3T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.38552135288200370376489952736, −10.50915520167614710674180318157, −9.582001784051365475706511804837, −8.299773788037443150440152030007, −7.83677180437177030988966977732, −6.30751875162328606271102464687, −4.93646751948018415383376413498, −4.03832076092577603264777613661, −1.59414625185452997779919612534, 0,
1.59414625185452997779919612534, 4.03832076092577603264777613661, 4.93646751948018415383376413498, 6.30751875162328606271102464687, 7.83677180437177030988966977732, 8.299773788037443150440152030007, 9.582001784051365475706511804837, 10.50915520167614710674180318157, 11.38552135288200370376489952736
