| L(s) = 1 | + 3-s − 5-s − 2·7-s − 2·9-s + 3·11-s + 13-s − 15-s + 8·17-s − 2·21-s + 4·23-s − 4·25-s − 5·27-s + 29-s − 3·31-s + 3·33-s + 2·35-s − 8·37-s + 39-s + 2·41-s + 11·43-s + 2·45-s + 13·47-s − 3·49-s + 8·51-s + 11·53-s − 3·55-s + 8·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.755·7-s − 2/3·9-s + 0.904·11-s + 0.277·13-s − 0.258·15-s + 1.94·17-s − 0.436·21-s + 0.834·23-s − 4/5·25-s − 0.962·27-s + 0.185·29-s − 0.538·31-s + 0.522·33-s + 0.338·35-s − 1.31·37-s + 0.160·39-s + 0.312·41-s + 1.67·43-s + 0.298·45-s + 1.89·47-s − 3/7·49-s + 1.12·51-s + 1.51·53-s − 0.404·55-s + 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.826470090\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.826470090\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 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
−9.187696200797261240168787461144, −8.538714587534864706883628474190, −7.68992355213504990402216682341, −7.02533007960674119090827223190, −5.97874250935484417499511635389, −5.35705266569945906014064710560, −3.80259981482385994331528030455, −3.53944000086915839540994955829, −2.44287708620897533980700790337, −0.907460307275556682567938804358,
0.907460307275556682567938804358, 2.44287708620897533980700790337, 3.53944000086915839540994955829, 3.80259981482385994331528030455, 5.35705266569945906014064710560, 5.97874250935484417499511635389, 7.02533007960674119090827223190, 7.68992355213504990402216682341, 8.538714587534864706883628474190, 9.187696200797261240168787461144
