| L(s) = 1 | − 2-s + 3·3-s + 4-s − 3·6-s − 8-s + 6·9-s − 2·11-s + 3·12-s + 16-s − 4·17-s − 6·18-s + 6·19-s + 2·22-s − 3·23-s − 3·24-s + 9·27-s + 9·29-s + 4·31-s − 32-s − 6·33-s + 4·34-s + 6·36-s + 4·37-s − 6·38-s + 7·41-s + 5·43-s − 2·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.73·3-s + 1/2·4-s − 1.22·6-s − 0.353·8-s + 2·9-s − 0.603·11-s + 0.866·12-s + 1/4·16-s − 0.970·17-s − 1.41·18-s + 1.37·19-s + 0.426·22-s − 0.625·23-s − 0.612·24-s + 1.73·27-s + 1.67·29-s + 0.718·31-s − 0.176·32-s − 1.04·33-s + 0.685·34-s + 36-s + 0.657·37-s − 0.973·38-s + 1.09·41-s + 0.762·43-s − 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.521089737\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.521089737\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.965481288250785898829283008574, −8.153276850381094005493949185211, −7.75482899372874136780437024042, −7.02697016707504450398481452546, −6.07072648860756642778838685688, −4.77487246169842344977010988717, −3.87001358016329068702122423338, −2.78588534104432571077405186020, −2.40510021455596195483136420469, −1.08922700890937652804724612626,
1.08922700890937652804724612626, 2.40510021455596195483136420469, 2.78588534104432571077405186020, 3.87001358016329068702122423338, 4.77487246169842344977010988717, 6.07072648860756642778838685688, 7.02697016707504450398481452546, 7.75482899372874136780437024042, 8.153276850381094005493949185211, 8.965481288250785898829283008574
