| L(s) = 1 | − 2-s − 0.481·3-s + 4-s + 5-s + 0.481·6-s − 1.48·7-s − 8-s − 2.76·9-s − 10-s + 11-s − 0.481·12-s + 6.63·13-s + 1.48·14-s − 0.481·15-s + 16-s + 1.13·17-s + 2.76·18-s − 5.73·19-s + 20-s + 0.712·21-s − 22-s + 6.96·23-s + 0.481·24-s + 25-s − 6.63·26-s + 2.77·27-s − 1.48·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.277·3-s + 0.5·4-s + 0.447·5-s + 0.196·6-s − 0.559·7-s − 0.353·8-s − 0.922·9-s − 0.316·10-s + 0.301·11-s − 0.138·12-s + 1.84·13-s + 0.395·14-s − 0.124·15-s + 0.250·16-s + 0.274·17-s + 0.652·18-s − 1.31·19-s + 0.223·20-s + 0.155·21-s − 0.213·22-s + 1.45·23-s + 0.0982·24-s + 0.200·25-s − 1.30·26-s + 0.534·27-s − 0.279·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.006804906\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.006804906\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| good | 3 | \( 1 + 0.481T + 3T^{2} \) |
| 7 | \( 1 + 1.48T + 7T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 - 6.63T + 13T^{2} \) |
| 17 | \( 1 - 1.13T + 17T^{2} \) |
| 19 | \( 1 + 5.73T + 19T^{2} \) |
| 23 | \( 1 - 6.96T + 23T^{2} \) |
| 29 | \( 1 - 4.63T + 29T^{2} \) |
| 31 | \( 1 - 1.67T + 31T^{2} \) |
| 37 | \( 1 - 8.59T + 37T^{2} \) |
| 41 | \( 1 + 4.79T + 41T^{2} \) |
| 43 | \( 1 + 6.60T + 43T^{2} \) |
| 47 | \( 1 - 4.06T + 47T^{2} \) |
| 53 | \( 1 - 9.86T + 53T^{2} \) |
| 59 | \( 1 - 10.5T + 59T^{2} \) |
| 61 | \( 1 - 9.37T + 61T^{2} \) |
| 71 | \( 1 + 0.413T + 71T^{2} \) |
| 73 | \( 1 - 10.2T + 73T^{2} \) |
| 79 | \( 1 - 6.05T + 79T^{2} \) |
| 83 | \( 1 - 13.5T + 83T^{2} \) |
| 89 | \( 1 + 3.31T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
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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
−10.58919888821588861688859671856, −9.592541343255556764239259405782, −8.674490025885868577688555742329, −8.310658417495142063763932085339, −6.66095781015900620893084408067, −6.34549542457982399067606360625, −5.31813788610048642752071723198, −3.74082551941033103234070644201, −2.61079144938469805715669072744, −0.996591454418429453617589627294,
0.996591454418429453617589627294, 2.61079144938469805715669072744, 3.74082551941033103234070644201, 5.31813788610048642752071723198, 6.34549542457982399067606360625, 6.66095781015900620893084408067, 8.310658417495142063763932085339, 8.674490025885868577688555742329, 9.592541343255556764239259405782, 10.58919888821588861688859671856
