| L(s) = 1 | + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s + 3·11-s + 2·13-s − 14-s + 16-s + 5·19-s − 20-s + 3·22-s − 9·23-s + 25-s + 2·26-s − 28-s + 31-s + 32-s + 35-s + 8·37-s + 5·38-s − 40-s + 11·43-s + 3·44-s − 9·46-s + 6·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s + 0.353·8-s − 0.316·10-s + 0.904·11-s + 0.554·13-s − 0.267·14-s + 1/4·16-s + 1.14·19-s − 0.223·20-s + 0.639·22-s − 1.87·23-s + 1/5·25-s + 0.392·26-s − 0.188·28-s + 0.179·31-s + 0.176·32-s + 0.169·35-s + 1.31·37-s + 0.811·38-s − 0.158·40-s + 1.67·43-s + 0.452·44-s − 1.32·46-s + 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.807744324\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.807744324\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−8.823296505213544891998302368019, −7.81274312976849185188659318694, −7.33591832206634595181516248511, −6.17548996042192493230446588331, −5.99431652998045200506912041969, −4.74811762028286937034304091577, −3.96280908569880185728912700218, −3.40120120676549713671210936822, −2.26954342882291794390735368364, −0.970760554217446015952289485453,
0.970760554217446015952289485453, 2.26954342882291794390735368364, 3.40120120676549713671210936822, 3.96280908569880185728912700218, 4.74811762028286937034304091577, 5.99431652998045200506912041969, 6.17548996042192493230446588331, 7.33591832206634595181516248511, 7.81274312976849185188659318694, 8.823296505213544891998302368019
