L(s) = 1 | − 3-s − 7-s − 2·9-s + 3·11-s + 13-s − 2·19-s + 21-s + 3·23-s − 5·25-s + 5·27-s − 5·31-s − 3·33-s − 7·37-s − 39-s + 3·41-s − 8·43-s + 3·47-s + 49-s − 12·53-s + 2·57-s − 6·59-s − 61-s + 2·63-s − 5·67-s − 3·69-s − 12·71-s + 11·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.377·7-s − 2/3·9-s + 0.904·11-s + 0.277·13-s − 0.458·19-s + 0.218·21-s + 0.625·23-s − 25-s + 0.962·27-s − 0.898·31-s − 0.522·33-s − 1.15·37-s − 0.160·39-s + 0.468·41-s − 1.21·43-s + 0.437·47-s + 1/7·49-s − 1.64·53-s + 0.264·57-s − 0.781·59-s − 0.128·61-s + 0.251·63-s − 0.610·67-s − 0.361·69-s − 1.42·71-s + 1.28·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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
−9.081078374331817625364448213995, −8.474366624001991302416802379216, −7.37908920927623601208890863309, −6.49972537294879181773432831310, −5.92908447901902110795424240562, −5.02890614745559905024763859198, −3.94823754743172236057368898190, −3.03438118997979637652891820124, −1.60256072097181437267558889267, 0,
1.60256072097181437267558889267, 3.03438118997979637652891820124, 3.94823754743172236057368898190, 5.02890614745559905024763859198, 5.92908447901902110795424240562, 6.49972537294879181773432831310, 7.37908920927623601208890863309, 8.474366624001991302416802379216, 9.081078374331817625364448213995