| L(s) = 1 | − 2-s − 3-s + 4-s − 3.15·5-s + 6-s − 4.69·7-s − 8-s + 9-s + 3.15·10-s + 0.137·11-s − 12-s + 4.69·14-s + 3.15·15-s + 16-s − 5.60·17-s − 18-s − 4.98·19-s − 3.15·20-s + 4.69·21-s − 0.137·22-s − 6.09·23-s + 24-s + 4.97·25-s − 27-s − 4.69·28-s − 0.850·29-s − 3.15·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.41·5-s + 0.408·6-s − 1.77·7-s − 0.353·8-s + 0.333·9-s + 0.998·10-s + 0.0413·11-s − 0.288·12-s + 1.25·14-s + 0.815·15-s + 0.250·16-s − 1.35·17-s − 0.235·18-s − 1.14·19-s − 0.706·20-s + 1.02·21-s − 0.0292·22-s − 1.27·23-s + 0.204·24-s + 0.995·25-s − 0.192·27-s − 0.886·28-s − 0.157·29-s − 0.576·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2111868477\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2111868477\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 3.15T + 5T^{2} \) |
| 7 | \( 1 + 4.69T + 7T^{2} \) |
| 11 | \( 1 - 0.137T + 11T^{2} \) |
| 17 | \( 1 + 5.60T + 17T^{2} \) |
| 19 | \( 1 + 4.98T + 19T^{2} \) |
| 23 | \( 1 + 6.09T + 23T^{2} \) |
| 29 | \( 1 + 0.850T + 29T^{2} \) |
| 31 | \( 1 - 6.23T + 31T^{2} \) |
| 37 | \( 1 - 11.7T + 37T^{2} \) |
| 41 | \( 1 + 4.27T + 41T^{2} \) |
| 43 | \( 1 + 2.09T + 43T^{2} \) |
| 47 | \( 1 - 4.98T + 47T^{2} \) |
| 53 | \( 1 + 1.82T + 53T^{2} \) |
| 59 | \( 1 + 5.89T + 59T^{2} \) |
| 61 | \( 1 - 4.39T + 61T^{2} \) |
| 67 | \( 1 + 4.71T + 67T^{2} \) |
| 71 | \( 1 + 0.0978T + 71T^{2} \) |
| 73 | \( 1 + 2.32T + 73T^{2} \) |
| 79 | \( 1 - 14.5T + 79T^{2} \) |
| 83 | \( 1 + 9.85T + 83T^{2} \) |
| 89 | \( 1 - 17.0T + 89T^{2} \) |
| 97 | \( 1 - 2.12T + 97T^{2} \) |
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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
−9.998534953463524196579445916915, −9.155021636823460135001196736797, −8.305230003868850733595675970769, −7.46107318351876880779526718114, −6.49878668382275200973814407429, −6.20080569636341307151379805717, −4.45343181693999678945633380812, −3.73555830132562197921293069075, −2.51521023013019121394191609947, −0.37852683374557655394310314207,
0.37852683374557655394310314207, 2.51521023013019121394191609947, 3.73555830132562197921293069075, 4.45343181693999678945633380812, 6.20080569636341307151379805717, 6.49878668382275200973814407429, 7.46107318351876880779526718114, 8.305230003868850733595675970769, 9.155021636823460135001196736797, 9.998534953463524196579445916915
