L(s) = 1 | − 1.93·2-s − 3-s + 1.73·4-s + 1.93·6-s + 0.517·8-s + 9-s − 3.46·11-s − 1.73·12-s + 4·13-s − 4.46·16-s + 4·17-s − 1.93·18-s − 0.378·19-s + 6.69·22-s + 6.31·23-s − 0.517·24-s − 7.72·26-s − 27-s + 8.92·29-s − 7.34·31-s + 7.58·32-s + 3.46·33-s − 7.72·34-s + 1.73·36-s − 0.757·37-s + 0.732·38-s − 4·39-s + ⋯ |
L(s) = 1 | − 1.36·2-s − 0.577·3-s + 0.866·4-s + 0.788·6-s + 0.183·8-s + 0.333·9-s − 1.04·11-s − 0.500·12-s + 1.10·13-s − 1.11·16-s + 0.970·17-s − 0.455·18-s − 0.0869·19-s + 1.42·22-s + 1.31·23-s − 0.105·24-s − 1.51·26-s − 0.192·27-s + 1.65·29-s − 1.31·31-s + 1.34·32-s + 0.603·33-s − 1.32·34-s + 0.288·36-s − 0.124·37-s + 0.118·38-s − 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6924791331\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6924791331\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 1.93T + 2T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 + 0.378T + 19T^{2} \) |
| 23 | \( 1 - 6.31T + 23T^{2} \) |
| 29 | \( 1 - 8.92T + 29T^{2} \) |
| 31 | \( 1 + 7.34T + 31T^{2} \) |
| 37 | \( 1 + 0.757T + 37T^{2} \) |
| 41 | \( 1 + 8.48T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 + 7.34T + 53T^{2} \) |
| 59 | \( 1 - 10.5T + 59T^{2} \) |
| 61 | \( 1 - 9.14T + 61T^{2} \) |
| 67 | \( 1 + 6.96T + 67T^{2} \) |
| 71 | \( 1 - 14.3T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + 4.14T + 89T^{2} \) |
| 97 | \( 1 - 5.07T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.394945360520378792850589384254, −8.090526636614039865064793708397, −7.12073134325742323519380875362, −6.63345348947785708821396102292, −5.51998244030968874708798296047, −4.99237937445917782148512611504, −3.82553557400490251686701075571, −2.75034245525125024705289118519, −1.51266737127042097706832080440, −0.66289297090853677245546156900,
0.66289297090853677245546156900, 1.51266737127042097706832080440, 2.75034245525125024705289118519, 3.82553557400490251686701075571, 4.99237937445917782148512611504, 5.51998244030968874708798296047, 6.63345348947785708821396102292, 7.12073134325742323519380875362, 8.090526636614039865064793708397, 8.394945360520378792850589384254