| L(s) = 1 | − 3-s + 7-s + 9-s + 2·13-s − 2·17-s − 21-s − 27-s − 6·29-s + 4·31-s − 2·37-s − 2·39-s + 10·41-s + 4·43-s + 49-s + 2·51-s + 2·53-s − 4·59-s − 6·61-s + 63-s − 12·67-s + 12·71-s + 2·73-s + 81-s + 4·83-s + 6·87-s + 10·89-s + 2·91-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.554·13-s − 0.485·17-s − 0.218·21-s − 0.192·27-s − 1.11·29-s + 0.718·31-s − 0.328·37-s − 0.320·39-s + 1.56·41-s + 0.609·43-s + 1/7·49-s + 0.280·51-s + 0.274·53-s − 0.520·59-s − 0.768·61-s + 0.125·63-s − 1.46·67-s + 1.42·71-s + 0.234·73-s + 1/9·81-s + 0.439·83-s + 0.643·87-s + 1.05·89-s + 0.209·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.755542645\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.755542645\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
| 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
−15.03990290585854, −14.55015221676124, −13.77127286785788, −13.54252309798128, −12.78289078916454, −12.39829087405665, −11.74588371115283, −11.27090002953423, −10.76505812220733, −10.44250862620242, −9.539149440124519, −9.176506846014177, −8.545109313192527, −7.804585499474077, −7.455034537389940, −6.653295949448224, −6.154313356152534, −5.611659030616229, −4.972502318146803, −4.309068367138710, −3.824932006850662, −2.930540213625946, −2.136073282338353, −1.380506818356430, −0.5417441179837895,
0.5417441179837895, 1.380506818356430, 2.136073282338353, 2.930540213625946, 3.824932006850662, 4.309068367138710, 4.972502318146803, 5.611659030616229, 6.154313356152534, 6.653295949448224, 7.455034537389940, 7.804585499474077, 8.545109313192527, 9.176506846014177, 9.539149440124519, 10.44250862620242, 10.76505812220733, 11.27090002953423, 11.74588371115283, 12.39829087405665, 12.78289078916454, 13.54252309798128, 13.77127286785788, 14.55015221676124, 15.03990290585854
