L(s) = 1 | + 3-s − 5-s + 7-s + 9-s + 2·11-s + 4·13-s − 15-s − 2·17-s − 6·19-s + 21-s + 4·23-s + 25-s + 27-s + 10·29-s − 2·31-s + 2·33-s − 35-s + 2·37-s + 4·39-s − 10·41-s + 4·43-s − 45-s + 8·47-s + 49-s − 2·51-s − 4·53-s − 2·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.603·11-s + 1.10·13-s − 0.258·15-s − 0.485·17-s − 1.37·19-s + 0.218·21-s + 0.834·23-s + 1/5·25-s + 0.192·27-s + 1.85·29-s − 0.359·31-s + 0.348·33-s − 0.169·35-s + 0.328·37-s + 0.640·39-s − 1.56·41-s + 0.609·43-s − 0.149·45-s + 1.16·47-s + 1/7·49-s − 0.280·51-s − 0.549·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.693049733\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.693049733\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−8.202895607397633921034815918042, −7.27253775033112195641491208651, −6.63531859643468812020134360532, −6.06167445052925301847722751273, −4.91405803997892797007361916701, −4.28814657639165362498759127403, −3.65681516562296852641553769336, −2.78478654312962465632596899291, −1.82765064622844198050247586509, −0.852165152175321061071138373084,
0.852165152175321061071138373084, 1.82765064622844198050247586509, 2.78478654312962465632596899291, 3.65681516562296852641553769336, 4.28814657639165362498759127403, 4.91405803997892797007361916701, 6.06167445052925301847722751273, 6.63531859643468812020134360532, 7.27253775033112195641491208651, 8.202895607397633921034815918042