| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s + 11-s + 2·13-s − 14-s + 16-s + 2·19-s − 22-s − 5·25-s − 2·26-s + 28-s + 6·29-s + 2·31-s − 32-s + 2·37-s − 2·38-s − 4·43-s + 44-s + 6·47-s + 49-s + 5·50-s + 2·52-s + 6·53-s − 56-s − 6·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s + 0.301·11-s + 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.458·19-s − 0.213·22-s − 25-s − 0.392·26-s + 0.188·28-s + 1.11·29-s + 0.359·31-s − 0.176·32-s + 0.328·37-s − 0.324·38-s − 0.609·43-s + 0.150·44-s + 0.875·47-s + 1/7·49-s + 0.707·50-s + 0.277·52-s + 0.824·53-s − 0.133·56-s − 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.313716839\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.313716839\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−9.575410322452751658082485313148, −8.707141431781925178766899910941, −8.104024357108211588654438142125, −7.27269450626182225527303312385, −6.40050362102582630929450786700, −5.57921544774938402027851071868, −4.44481069307592434914952580537, −3.37789225057720607500704079136, −2.16035280583295696954900337452, −0.960161767014088955334438808530,
0.960161767014088955334438808530, 2.16035280583295696954900337452, 3.37789225057720607500704079136, 4.44481069307592434914952580537, 5.57921544774938402027851071868, 6.40050362102582630929450786700, 7.27269450626182225527303312385, 8.104024357108211588654438142125, 8.707141431781925178766899910941, 9.575410322452751658082485313148
