| L(s) = 1 | − 2-s + 3·3-s − 4-s + 5-s − 3·6-s + 7-s + 3·8-s + 6·9-s − 10-s − 3·12-s − 2·13-s − 14-s + 3·15-s − 16-s + 17-s − 6·18-s + 5·19-s − 20-s + 3·21-s + 4·23-s + 9·24-s − 4·25-s + 2·26-s + 9·27-s − 28-s − 9·29-s − 3·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.73·3-s − 1/2·4-s + 0.447·5-s − 1.22·6-s + 0.377·7-s + 1.06·8-s + 2·9-s − 0.316·10-s − 0.866·12-s − 0.554·13-s − 0.267·14-s + 0.774·15-s − 1/4·16-s + 0.242·17-s − 1.41·18-s + 1.14·19-s − 0.223·20-s + 0.654·21-s + 0.834·23-s + 1.83·24-s − 4/5·25-s + 0.392·26-s + 1.73·27-s − 0.188·28-s − 1.67·29-s − 0.547·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.055252542\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.055252542\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−9.820300756363784788156777981582, −9.018917109557403069799404864127, −8.565219910847943109455452716265, −7.55601681782054043840667679343, −7.31253152802736913256568221602, −5.53924369319540680170288259197, −4.50118603566072485034914834021, −3.52089292641192492900074825694, −2.40115404328902331276645060404, −1.32502079542108420499627702451,
1.32502079542108420499627702451, 2.40115404328902331276645060404, 3.52089292641192492900074825694, 4.50118603566072485034914834021, 5.53924369319540680170288259197, 7.31253152802736913256568221602, 7.55601681782054043840667679343, 8.565219910847943109455452716265, 9.018917109557403069799404864127, 9.820300756363784788156777981582
