L(s) = 1 | − 2-s + 4-s − 2·7-s − 8-s + 3·11-s − 5·13-s + 2·14-s + 16-s + 3·17-s − 4·19-s − 3·22-s + 9·23-s + 5·26-s − 2·28-s − 3·29-s + 5·31-s − 32-s − 3·34-s + 10·37-s + 4·38-s + 43-s + 3·44-s − 9·46-s − 9·47-s − 3·49-s − 5·52-s + 12·53-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s + 0.904·11-s − 1.38·13-s + 0.534·14-s + 1/4·16-s + 0.727·17-s − 0.917·19-s − 0.639·22-s + 1.87·23-s + 0.980·26-s − 0.377·28-s − 0.557·29-s + 0.898·31-s − 0.176·32-s − 0.514·34-s + 1.64·37-s + 0.648·38-s + 0.152·43-s + 0.452·44-s − 1.32·46-s − 1.31·47-s − 3/7·49-s − 0.693·52-s + 1.64·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.041098047\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.041098047\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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
−9.707858034848317020362067475421, −8.922579168856168548524542452123, −8.070256847202258401910502361454, −7.06033652698638292177384238301, −6.64584491604697513330173390466, −5.57101597774882988647982419305, −4.47711773079826783875439923792, −3.28861746879212715367092171206, −2.33640644197085955440984438866, −0.821320521629176164137238639055,
0.821320521629176164137238639055, 2.33640644197085955440984438866, 3.28861746879212715367092171206, 4.47711773079826783875439923792, 5.57101597774882988647982419305, 6.64584491604697513330173390466, 7.06033652698638292177384238301, 8.070256847202258401910502361454, 8.922579168856168548524542452123, 9.707858034848317020362067475421