L(s) = 1 | + 3-s + 2·5-s + 9-s + 13-s + 2·15-s + 2·17-s − 4·19-s − 25-s + 27-s + 6·29-s − 2·37-s + 39-s + 6·41-s − 12·43-s + 2·45-s − 4·47-s − 7·49-s + 2·51-s + 6·53-s − 4·57-s − 8·59-s − 2·61-s + 2·65-s + 4·67-s − 12·71-s − 14·73-s − 75-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1/3·9-s + 0.277·13-s + 0.516·15-s + 0.485·17-s − 0.917·19-s − 1/5·25-s + 0.192·27-s + 1.11·29-s − 0.328·37-s + 0.160·39-s + 0.937·41-s − 1.82·43-s + 0.298·45-s − 0.583·47-s − 49-s + 0.280·51-s + 0.824·53-s − 0.529·57-s − 1.04·59-s − 0.256·61-s + 0.248·65-s + 0.488·67-s − 1.42·71-s − 1.63·73-s − 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.777686788\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.777686788\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 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
−11.71381507191045566206980370613, −10.48713524691747743427093183397, −9.843379801158719744890276376324, −8.860624431335414724430887662727, −8.023927242894012661962773978846, −6.75712783178321908907460309814, −5.82101553029460348251109404035, −4.52332315712058664242014926650, −3.09863617379496380583398157204, −1.76107356520720300622704659457,
1.76107356520720300622704659457, 3.09863617379496380583398157204, 4.52332315712058664242014926650, 5.82101553029460348251109404035, 6.75712783178321908907460309814, 8.023927242894012661962773978846, 8.860624431335414724430887662727, 9.843379801158719744890276376324, 10.48713524691747743427093183397, 11.71381507191045566206980370613