| L(s) = 1 | − 3-s + 0.618·5-s + 7-s + 9-s + 3·11-s − 1.61·13-s − 0.618·15-s − 6.70·17-s − 0.236·19-s − 21-s + 23-s − 4.61·25-s − 27-s − 6.23·29-s + 3·31-s − 3·33-s + 0.618·35-s + 0.527·37-s + 1.61·39-s + 5.94·41-s − 2.09·43-s + 0.618·45-s + 4.76·47-s + 49-s + 6.70·51-s + 6.32·53-s + 1.85·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.276·5-s + 0.377·7-s + 0.333·9-s + 0.904·11-s − 0.448·13-s − 0.159·15-s − 1.62·17-s − 0.0541·19-s − 0.218·21-s + 0.208·23-s − 0.923·25-s − 0.192·27-s − 1.15·29-s + 0.538·31-s − 0.522·33-s + 0.104·35-s + 0.0867·37-s + 0.259·39-s + 0.928·41-s − 0.318·43-s + 0.0921·45-s + 0.694·47-s + 0.142·49-s + 0.939·51-s + 0.868·53-s + 0.250·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.593872952\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.593872952\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - 0.618T + 5T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + 1.61T + 13T^{2} \) |
| 17 | \( 1 + 6.70T + 17T^{2} \) |
| 19 | \( 1 + 0.236T + 19T^{2} \) |
| 29 | \( 1 + 6.23T + 29T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 - 0.527T + 37T^{2} \) |
| 41 | \( 1 - 5.94T + 41T^{2} \) |
| 43 | \( 1 + 2.09T + 43T^{2} \) |
| 47 | \( 1 - 4.76T + 47T^{2} \) |
| 53 | \( 1 - 6.32T + 53T^{2} \) |
| 59 | \( 1 - 7.38T + 59T^{2} \) |
| 61 | \( 1 + 0.854T + 61T^{2} \) |
| 67 | \( 1 - 13.5T + 67T^{2} \) |
| 71 | \( 1 - 2.38T + 71T^{2} \) |
| 73 | \( 1 - 6.23T + 73T^{2} \) |
| 79 | \( 1 + 6.70T + 79T^{2} \) |
| 83 | \( 1 + 1.47T + 83T^{2} \) |
| 89 | \( 1 - 7.79T + 89T^{2} \) |
| 97 | \( 1 - 14.4T + 97T^{2} \) |
| 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
−7.76510697465294518208095151697, −7.03215543771166556290692087471, −6.50136686595623507719007010167, −5.79502308844506216933928996637, −5.10040932924411161093909520212, −4.29294202007692834825117800997, −3.80084775561380021621834420549, −2.44175164493076720499479069033, −1.82154885748670611987139882791, −0.64602063457949569871679077669,
0.64602063457949569871679077669, 1.82154885748670611987139882791, 2.44175164493076720499479069033, 3.80084775561380021621834420549, 4.29294202007692834825117800997, 5.10040932924411161093909520212, 5.79502308844506216933928996637, 6.50136686595623507719007010167, 7.03215543771166556290692087471, 7.76510697465294518208095151697
