| L(s) = 1 | + 3-s + 3.11·5-s + 4.35·7-s + 9-s + 11-s + 13-s + 3.11·15-s + 2.60·17-s − 4.35·19-s + 4.35·21-s − 0.513·23-s + 4.72·25-s + 27-s − 3.47·29-s + 7.58·31-s + 33-s + 13.5·35-s − 1.83·37-s + 39-s + 7.37·41-s − 10.3·43-s + 3.11·45-s − 2.62·47-s + 11.9·49-s + 2.60·51-s − 8.94·53-s + 3.11·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.39·5-s + 1.64·7-s + 0.333·9-s + 0.301·11-s + 0.277·13-s + 0.805·15-s + 0.631·17-s − 0.998·19-s + 0.949·21-s − 0.107·23-s + 0.944·25-s + 0.192·27-s − 0.644·29-s + 1.36·31-s + 0.174·33-s + 2.29·35-s − 0.302·37-s + 0.160·39-s + 1.15·41-s − 1.57·43-s + 0.464·45-s − 0.383·47-s + 1.70·49-s + 0.364·51-s − 1.22·53-s + 0.420·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.558411779\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.558411779\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 3.11T + 5T^{2} \) |
| 7 | \( 1 - 4.35T + 7T^{2} \) |
| 17 | \( 1 - 2.60T + 17T^{2} \) |
| 19 | \( 1 + 4.35T + 19T^{2} \) |
| 23 | \( 1 + 0.513T + 23T^{2} \) |
| 29 | \( 1 + 3.47T + 29T^{2} \) |
| 31 | \( 1 - 7.58T + 31T^{2} \) |
| 37 | \( 1 + 1.83T + 37T^{2} \) |
| 41 | \( 1 - 7.37T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 + 2.62T + 47T^{2} \) |
| 53 | \( 1 + 8.94T + 53T^{2} \) |
| 59 | \( 1 - 3.49T + 59T^{2} \) |
| 61 | \( 1 - 4.39T + 61T^{2} \) |
| 67 | \( 1 + 8.79T + 67T^{2} \) |
| 71 | \( 1 + 3.83T + 71T^{2} \) |
| 73 | \( 1 - 7.56T + 73T^{2} \) |
| 79 | \( 1 - 5.63T + 79T^{2} \) |
| 83 | \( 1 - 3.49T + 83T^{2} \) |
| 89 | \( 1 + 2.72T + 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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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
−8.157669437668024716021888591470, −7.38163905539552232910391642802, −6.45073879287441279809923732981, −5.89486732812692941975325915579, −5.04328325226553345126918097420, −4.53439211951609154892628398948, −3.54882856458733147604858191113, −2.45261683822311320517056215900, −1.83553146030323635442579387324, −1.20731949929279831187709218048,
1.20731949929279831187709218048, 1.83553146030323635442579387324, 2.45261683822311320517056215900, 3.54882856458733147604858191113, 4.53439211951609154892628398948, 5.04328325226553345126918097420, 5.89486732812692941975325915579, 6.45073879287441279809923732981, 7.38163905539552232910391642802, 8.157669437668024716021888591470
