| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s − 4·13-s + 15-s − 2·19-s + 21-s − 23-s + 25-s − 27-s + 6·29-s − 2·31-s + 35-s − 10·37-s + 4·39-s + 6·41-s + 4·43-s − 45-s − 6·47-s + 49-s − 6·53-s + 2·57-s − 12·59-s − 10·61-s − 63-s + 4·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.10·13-s + 0.258·15-s − 0.458·19-s + 0.218·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 0.359·31-s + 0.169·35-s − 1.64·37-s + 0.640·39-s + 0.937·41-s + 0.609·43-s − 0.149·45-s − 0.875·47-s + 1/7·49-s − 0.824·53-s + 0.264·57-s − 1.56·59-s − 1.28·61-s − 0.125·63-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5032290657\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5032290657\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−14.88987300060495, −14.17418416777912, −13.99487050081815, −13.07731041044436, −12.57227580537729, −12.26025014861563, −11.83418772591279, −11.05551246024644, −10.72485477874790, −10.09500288543324, −9.620699707776594, −9.014226455580490, −8.401199211146469, −7.674654793479019, −7.329118820096355, −6.574147647629952, −6.241204794186726, −5.425489132717972, −4.827063279507299, −4.402336437017189, −3.603998681042547, −2.949396522748645, −2.201536921377789, −1.329300235107905, −0.2767077199927733,
0.2767077199927733, 1.329300235107905, 2.201536921377789, 2.949396522748645, 3.603998681042547, 4.402336437017189, 4.827063279507299, 5.425489132717972, 6.241204794186726, 6.574147647629952, 7.329118820096355, 7.674654793479019, 8.401199211146469, 9.014226455580490, 9.620699707776594, 10.09500288543324, 10.72485477874790, 11.05551246024644, 11.83418772591279, 12.26025014861563, 12.57227580537729, 13.07731041044436, 13.99487050081815, 14.17418416777912, 14.88987300060495
