L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 7-s − 8-s + 9-s + 3·11-s + 12-s − 4·13-s + 14-s + 16-s + 17-s − 18-s − 19-s − 21-s − 3·22-s − 24-s + 4·26-s + 27-s − 28-s + 5·31-s − 32-s + 3·33-s − 34-s + 36-s + 11·37-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.904·11-s + 0.288·12-s − 1.10·13-s + 0.267·14-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.229·19-s − 0.218·21-s − 0.639·22-s − 0.204·24-s + 0.784·26-s + 0.192·27-s − 0.188·28-s + 0.898·31-s − 0.176·32-s + 0.522·33-s − 0.171·34-s + 1/6·36-s + 1.80·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.539760202\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.539760202\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 17 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 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
−9.021999316713847773817045763581, −8.104830002642170843202267280067, −7.58567155663321117364764389655, −6.69904985354610417248101175339, −6.12061026907304874645619553829, −4.87542607194081224207784987705, −3.96907025798419538882327656342, −2.94668019400102269454902228743, −2.14101870925698585444088242163, −0.853446755779258579396256509190,
0.853446755779258579396256509190, 2.14101870925698585444088242163, 2.94668019400102269454902228743, 3.96907025798419538882327656342, 4.87542607194081224207784987705, 6.12061026907304874645619553829, 6.69904985354610417248101175339, 7.58567155663321117364764389655, 8.104830002642170843202267280067, 9.021999316713847773817045763581