L(s) = 1 | + 2-s + 3-s + 4-s − 3·5-s + 6-s − 3·7-s + 8-s + 9-s − 3·10-s + 12-s − 6·13-s − 3·14-s − 3·15-s + 16-s − 2·17-s + 18-s − 2·19-s − 3·20-s − 3·21-s − 6·23-s + 24-s + 4·25-s − 6·26-s + 27-s − 3·28-s + 4·29-s − 3·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s − 1.13·7-s + 0.353·8-s + 1/3·9-s − 0.948·10-s + 0.288·12-s − 1.66·13-s − 0.801·14-s − 0.774·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.458·19-s − 0.670·20-s − 0.654·21-s − 1.25·23-s + 0.204·24-s + 4/5·25-s − 1.17·26-s + 0.192·27-s − 0.566·28-s + 0.742·29-s − 0.547·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 + T \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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
−9.739576030462600961618355652819, −8.536902746450106053470154804549, −7.79040103804927106665258431984, −7.02028623688610468929627737027, −6.29749636831852881000301760966, −4.85301756910450646290943185771, −4.12973458050825905070728226137, −3.25758624328345452249415811059, −2.37775787775376028544956538733, 0,
2.37775787775376028544956538733, 3.25758624328345452249415811059, 4.12973458050825905070728226137, 4.85301756910450646290943185771, 6.29749636831852881000301760966, 7.02028623688610468929627737027, 7.79040103804927106665258431984, 8.536902746450106053470154804549, 9.739576030462600961618355652819