| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 3·7-s − 8-s + 9-s − 10-s − 11-s − 12-s + 3·14-s − 15-s + 16-s + 17-s − 18-s + 20-s + 3·21-s + 22-s + 23-s + 24-s − 4·25-s − 27-s − 3·28-s − 3·29-s + 30-s + 2·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.13·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.301·11-s − 0.288·12-s + 0.801·14-s − 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 0.223·20-s + 0.654·21-s + 0.213·22-s + 0.208·23-s + 0.204·24-s − 4/5·25-s − 0.192·27-s − 0.566·28-s − 0.557·29-s + 0.182·30-s + 0.359·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7288733142\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7288733142\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 13 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−13.02163547496275, −12.76215035519499, −11.95267003207790, −11.78542933624724, −11.15905201710227, −10.65774196143770, −10.17564417185493, −9.809391183068650, −9.494515229763162, −8.957928705567018, −8.416945673323705, −7.674034159944294, −7.500049486789970, −6.680285245073043, −6.408008847328063, −5.955262958949695, −5.446539501067803, −4.889781646428310, −4.176046674717635, −3.469843728353699, −3.078139212225379, −2.304224670503584, −1.809904843653614, −0.9911005130197885, −0.3249071780500154,
0.3249071780500154, 0.9911005130197885, 1.809904843653614, 2.304224670503584, 3.078139212225379, 3.469843728353699, 4.176046674717635, 4.889781646428310, 5.446539501067803, 5.955262958949695, 6.408008847328063, 6.680285245073043, 7.500049486789970, 7.674034159944294, 8.416945673323705, 8.957928705567018, 9.494515229763162, 9.809391183068650, 10.17564417185493, 10.65774196143770, 11.15905201710227, 11.78542933624724, 11.95267003207790, 12.76215035519499, 13.02163547496275
