L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.500 − 0.866i)4-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)7-s − 3·8-s − 0.999·10-s + (−1 − 1.73i)11-s + (2.5 − 4.33i)13-s + (0.499 − 0.866i)14-s + (0.500 + 0.866i)16-s − 3·17-s − 2·19-s + (−0.499 − 0.866i)20-s + (−0.999 + 1.73i)22-s + (3 − 5.19i)23-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.250 − 0.433i)4-s + (0.223 − 0.387i)5-s + (0.188 + 0.327i)7-s − 1.06·8-s − 0.316·10-s + (−0.301 − 0.522i)11-s + (0.693 − 1.20i)13-s + (0.133 − 0.231i)14-s + (0.125 + 0.216i)16-s − 0.727·17-s − 0.458·19-s + (−0.111 − 0.193i)20-s + (−0.213 + 0.369i)22-s + (0.625 − 1.08i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.394989 - 1.08522i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.394989 - 1.08522i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
good | 2 | \( 1 + (0.5 + 0.866i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (-0.5 + 0.866i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.5 + 4.33i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.5 + 4.33i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3 + 5.19i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 3T + 37T^{2} \) |
| 41 | \( 1 + (5 - 8.66i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.5 + 6.06i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1 + 1.73i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 15T + 73T^{2} \) |
| 79 | \( 1 + (7 + 12.1i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-9 - 15.5i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 5T + 89T^{2} \) |
| 97 | \( 1 + (-9 - 15.5i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.55897767049880909438734609177, −9.618407579326864730340446068605, −8.736711920879180341081651000729, −8.099026502834212597792292554655, −6.56789125892463257362783011090, −5.81639880377413204701500250239, −4.87833660444697249448705145547, −3.26482364491044152393212575235, −2.18063501557300371070918799665, −0.72607937281851104768554051954,
1.97431014771842500510714112996, 3.33811106840433212727430464719, 4.51345672418979712140820087402, 5.87369490167671048888794752659, 6.97723493604944721613105666151, 7.15464650050211429420095066676, 8.571085444848608780991941798973, 8.974806352255844406393976418875, 10.21160615803647056654168671298, 11.05978706743069858072622051843