L(s) = 1 | − 2-s + 3-s − 4-s − 5-s − 6-s − 5·7-s + 3·8-s + 9-s + 10-s − 4·11-s − 12-s − 4·13-s + 5·14-s − 15-s − 16-s + 6·17-s − 18-s − 2·19-s + 20-s − 5·21-s + 4·22-s − 3·23-s + 3·24-s − 4·25-s + 4·26-s + 27-s + 5·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s − 0.408·6-s − 1.88·7-s + 1.06·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s − 0.288·12-s − 1.10·13-s + 1.33·14-s − 0.258·15-s − 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.458·19-s + 0.223·20-s − 1.09·21-s + 0.852·22-s − 0.625·23-s + 0.612·24-s − 4/5·25-s + 0.784·26-s + 0.192·27-s + 0.944·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 67 | \( 1 - T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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
−12.28206918140350415129173431897, −10.37798583319944611001144167620, −9.925959776430250019913820005716, −9.140092761973951053457697213341, −7.908234279254522837888093498525, −7.29911852530154644846482005425, −5.65568854079860499573484952378, −4.05781959200863859195085231262, −2.81339700394220571358367305061, 0,
2.81339700394220571358367305061, 4.05781959200863859195085231262, 5.65568854079860499573484952378, 7.29911852530154644846482005425, 7.908234279254522837888093498525, 9.140092761973951053457697213341, 9.925959776430250019913820005716, 10.37798583319944611001144167620, 12.28206918140350415129173431897