L(s) = 1 | − 2-s + 0.567·3-s + 4-s − 2.51·5-s − 0.567·6-s + 3.57·7-s − 8-s − 2.67·9-s + 2.51·10-s − 5.90·11-s + 0.567·12-s − 3.72·13-s − 3.57·14-s − 1.42·15-s + 16-s + 5.80·17-s + 2.67·18-s − 1.27·19-s − 2.51·20-s + 2.02·21-s + 5.90·22-s − 3.04·23-s − 0.567·24-s + 1.31·25-s + 3.72·26-s − 3.22·27-s + 3.57·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.327·3-s + 0.5·4-s − 1.12·5-s − 0.231·6-s + 1.34·7-s − 0.353·8-s − 0.892·9-s + 0.794·10-s − 1.77·11-s + 0.163·12-s − 1.03·13-s − 0.954·14-s − 0.368·15-s + 0.250·16-s + 1.40·17-s + 0.631·18-s − 0.292·19-s − 0.562·20-s + 0.442·21-s + 1.25·22-s − 0.634·23-s − 0.115·24-s + 0.263·25-s + 0.729·26-s − 0.620·27-s + 0.674·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 502 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 502 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 251 | \( 1 + T \) |
good | 3 | \( 1 - 0.567T + 3T^{2} \) |
| 5 | \( 1 + 2.51T + 5T^{2} \) |
| 7 | \( 1 - 3.57T + 7T^{2} \) |
| 11 | \( 1 + 5.90T + 11T^{2} \) |
| 13 | \( 1 + 3.72T + 13T^{2} \) |
| 17 | \( 1 - 5.80T + 17T^{2} \) |
| 19 | \( 1 + 1.27T + 19T^{2} \) |
| 23 | \( 1 + 3.04T + 23T^{2} \) |
| 29 | \( 1 + 3.75T + 29T^{2} \) |
| 31 | \( 1 + 7.64T + 31T^{2} \) |
| 37 | \( 1 - 11.9T + 37T^{2} \) |
| 41 | \( 1 + 9.65T + 41T^{2} \) |
| 43 | \( 1 + 1.48T + 43T^{2} \) |
| 47 | \( 1 - 0.516T + 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 - 0.696T + 59T^{2} \) |
| 61 | \( 1 + 4.04T + 61T^{2} \) |
| 67 | \( 1 + 8.27T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 - 0.498T + 73T^{2} \) |
| 79 | \( 1 - 3.99T + 79T^{2} \) |
| 83 | \( 1 - 2.57T + 83T^{2} \) |
| 89 | \( 1 - 10.1T + 89T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
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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
−10.57190833564405078407634006740, −9.558109282953204223679706956184, −8.314902038684949731412513514290, −7.78806179135504803880014915181, −7.59431627252532567400669252244, −5.66339252628662261449680487868, −4.83332722381222585830626258550, −3.32581578103804794736852509681, −2.15023419478749293633621883610, 0,
2.15023419478749293633621883610, 3.32581578103804794736852509681, 4.83332722381222585830626258550, 5.66339252628662261449680487868, 7.59431627252532567400669252244, 7.78806179135504803880014915181, 8.314902038684949731412513514290, 9.558109282953204223679706956184, 10.57190833564405078407634006740