| L(s) = 1 | − 2-s − 1.34·3-s + 4-s + 5-s + 1.34·6-s − 7-s − 8-s − 1.19·9-s − 10-s − 1.34·12-s + 4.31·13-s + 14-s − 1.34·15-s + 16-s + 2.63·17-s + 1.19·18-s + 6.54·19-s + 20-s + 1.34·21-s − 3.16·23-s + 1.34·24-s + 25-s − 4.31·26-s + 5.63·27-s − 28-s + 1.13·29-s + 1.34·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.775·3-s + 0.5·4-s + 0.447·5-s + 0.548·6-s − 0.377·7-s − 0.353·8-s − 0.398·9-s − 0.316·10-s − 0.387·12-s + 1.19·13-s + 0.267·14-s − 0.346·15-s + 0.250·16-s + 0.639·17-s + 0.282·18-s + 1.50·19-s + 0.223·20-s + 0.293·21-s − 0.659·23-s + 0.274·24-s + 0.200·25-s − 0.845·26-s + 1.08·27-s − 0.188·28-s + 0.210·29-s + 0.245·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.208603291\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.208603291\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 1.34T + 3T^{2} \) |
| 13 | \( 1 - 4.31T + 13T^{2} \) |
| 17 | \( 1 - 2.63T + 17T^{2} \) |
| 19 | \( 1 - 6.54T + 19T^{2} \) |
| 23 | \( 1 + 3.16T + 23T^{2} \) |
| 29 | \( 1 - 1.13T + 29T^{2} \) |
| 31 | \( 1 - 1.14T + 31T^{2} \) |
| 37 | \( 1 - 3.13T + 37T^{2} \) |
| 41 | \( 1 + 0.521T + 41T^{2} \) |
| 43 | \( 1 - 4.96T + 43T^{2} \) |
| 47 | \( 1 - 11.8T + 47T^{2} \) |
| 53 | \( 1 - 2.59T + 53T^{2} \) |
| 59 | \( 1 - 10.9T + 59T^{2} \) |
| 61 | \( 1 + 5.82T + 61T^{2} \) |
| 67 | \( 1 + 9.72T + 67T^{2} \) |
| 71 | \( 1 + 4.50T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 + 17.0T + 79T^{2} \) |
| 83 | \( 1 + 9.06T + 83T^{2} \) |
| 89 | \( 1 - 7.17T + 89T^{2} \) |
| 97 | \( 1 - 9.85T + 97T^{2} \) |
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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
−7.68186820352631907168107319058, −7.19480825762760144739330982008, −6.19549640539925602644736571988, −5.89560972800211336305629123262, −5.38161120049098064390405453326, −4.26868476080650823643260190627, −3.29806599120527543310827039467, −2.63526495533224471692499593273, −1.38653864844748541891632936359, −0.68878805653660445070752402389,
0.68878805653660445070752402389, 1.38653864844748541891632936359, 2.63526495533224471692499593273, 3.29806599120527543310827039467, 4.26868476080650823643260190627, 5.38161120049098064390405453326, 5.89560972800211336305629123262, 6.19549640539925602644736571988, 7.19480825762760144739330982008, 7.68186820352631907168107319058
