| L(s) = 1 | + 2-s + 3-s + 4-s + 3.56·5-s + 6-s − 7-s + 8-s + 9-s + 3.56·10-s − 1.56·11-s + 12-s + 13-s − 14-s + 3.56·15-s + 16-s − 6.68·17-s + 18-s − 4.68·19-s + 3.56·20-s − 21-s − 1.56·22-s − 5.56·23-s + 24-s + 7.68·25-s + 26-s + 27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.59·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s + 1.12·10-s − 0.470·11-s + 0.288·12-s + 0.277·13-s − 0.267·14-s + 0.919·15-s + 0.250·16-s − 1.62·17-s + 0.235·18-s − 1.07·19-s + 0.796·20-s − 0.218·21-s − 0.332·22-s − 1.15·23-s + 0.204·24-s + 1.53·25-s + 0.196·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.066361583\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.066361583\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 3.56T + 5T^{2} \) |
| 11 | \( 1 + 1.56T + 11T^{2} \) |
| 17 | \( 1 + 6.68T + 17T^{2} \) |
| 19 | \( 1 + 4.68T + 19T^{2} \) |
| 23 | \( 1 + 5.56T + 23T^{2} \) |
| 29 | \( 1 - 6.68T + 29T^{2} \) |
| 31 | \( 1 - 6.24T + 31T^{2} \) |
| 37 | \( 1 + 7.56T + 37T^{2} \) |
| 41 | \( 1 + 1.12T + 41T^{2} \) |
| 43 | \( 1 + 6.43T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 - 2.24T + 59T^{2} \) |
| 61 | \( 1 - 6.68T + 61T^{2} \) |
| 67 | \( 1 + 7.12T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 3.56T + 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 - 8.87T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 14.4T + 97T^{2} \) |
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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
−10.41857078250021618048528429289, −10.22060041031839152001460924305, −9.010485694780640031278337377431, −8.305995732921860230626379067102, −6.71997963436205347221010032644, −6.32928801133355306306372815819, −5.20040415974629017633015291518, −4.13517339401693938788755970479, −2.67685834892058680739712973755, −1.96532787879004213571976830321,
1.96532787879004213571976830321, 2.67685834892058680739712973755, 4.13517339401693938788755970479, 5.20040415974629017633015291518, 6.32928801133355306306372815819, 6.71997963436205347221010032644, 8.305995732921860230626379067102, 9.010485694780640031278337377431, 10.22060041031839152001460924305, 10.41857078250021618048528429289
