L(s) = 1 | − 2-s + 4-s − 8-s − 3·9-s + 4·11-s + 2·13-s + 16-s − 2·17-s + 3·18-s − 2·19-s − 4·22-s + 8·23-s − 5·25-s − 2·26-s + 8·31-s − 32-s + 2·34-s − 3·36-s + 8·37-s + 2·38-s + 6·41-s + 4·43-s + 4·44-s − 8·46-s − 8·47-s − 7·49-s + 5·50-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 9-s + 1.20·11-s + 0.554·13-s + 1/4·16-s − 0.485·17-s + 0.707·18-s − 0.458·19-s − 0.852·22-s + 1.66·23-s − 25-s − 0.392·26-s + 1.43·31-s − 0.176·32-s + 0.342·34-s − 1/2·36-s + 1.31·37-s + 0.324·38-s + 0.937·41-s + 0.609·43-s + 0.603·44-s − 1.17·46-s − 1.16·47-s − 49-s + 0.707·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.138501340\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.138501340\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 503 | \( 1 - T \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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
−9.776853377242166714321641509123, −9.045713520834134029024985085465, −8.518177580053212777301486775367, −7.60149755922095367584842945485, −6.48129940793107102458025933478, −6.05310818168592297446874013690, −4.69049447172712355583991760980, −3.51782401576178792422064914327, −2.39892043457039317717883491800, −0.946257334458773807700757924424,
0.946257334458773807700757924424, 2.39892043457039317717883491800, 3.51782401576178792422064914327, 4.69049447172712355583991760980, 6.05310818168592297446874013690, 6.48129940793107102458025933478, 7.60149755922095367584842945485, 8.518177580053212777301486775367, 9.045713520834134029024985085465, 9.776853377242166714321641509123