| L(s) = 1 | + 1.04·2-s + 3·3-s − 6.91·4-s − 6.71·5-s + 3.13·6-s + 8.77·7-s − 15.5·8-s + 9·9-s − 7.00·10-s + 17.1·11-s − 20.7·12-s − 57.2·13-s + 9.15·14-s − 20.1·15-s + 39.0·16-s − 103.·17-s + 9.39·18-s − 147.·19-s + 46.4·20-s + 26.3·21-s + 17.9·22-s − 114.·23-s − 46.6·24-s − 79.9·25-s − 59.7·26-s + 27·27-s − 60.6·28-s + ⋯ |
| L(s) = 1 | + 0.368·2-s + 0.577·3-s − 0.863·4-s − 0.600·5-s + 0.212·6-s + 0.473·7-s − 0.687·8-s + 0.333·9-s − 0.221·10-s + 0.471·11-s − 0.498·12-s − 1.22·13-s + 0.174·14-s − 0.346·15-s + 0.610·16-s − 1.47·17-s + 0.122·18-s − 1.78·19-s + 0.518·20-s + 0.273·21-s + 0.173·22-s − 1.03·23-s − 0.396·24-s − 0.639·25-s − 0.450·26-s + 0.192·27-s − 0.409·28-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 | \( 1 - 1.04T + 8T^{2} \) |
| 5 | \( 1 + 6.71T + 125T^{2} \) |
| 7 | \( 1 - 8.77T + 343T^{2} \) |
| 11 | \( 1 - 17.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 57.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 103.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 147.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 114.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 24.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 161.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 97.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 13.1T + 6.89e4T^{2} \) |
| 43 | \( 1 - 333.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 582.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 231.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 311.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 520.T + 2.26e5T^{2} \) |
| 71 | \( 1 - 62.1T + 3.57e5T^{2} \) |
| 73 | \( 1 + 894.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.03e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 801.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.52e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 320.T + 9.12e5T^{2} \) |
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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
−11.75868045075874394781955491975, −10.47352499035644908864176483544, −9.322401630325069169014084189015, −8.528253479861613863059367735660, −7.63076212982687791881688457860, −6.21897198183352610820356630677, −4.54311359391929379066814004795, −4.10903919175663666728746486171, −2.33958587247743485164368162051, 0,
2.33958587247743485164368162051, 4.10903919175663666728746486171, 4.54311359391929379066814004795, 6.21897198183352610820356630677, 7.63076212982687791881688457860, 8.528253479861613863059367735660, 9.322401630325069169014084189015, 10.47352499035644908864176483544, 11.75868045075874394781955491975
