L(s) = 1 | − 3·3-s − 15.3·5-s − 7·7-s + 9·9-s − 17.9·11-s + 17.3·13-s + 46.1·15-s − 84.3·17-s − 159.·19-s + 21·21-s + 115.·23-s + 111.·25-s − 27·27-s − 215.·29-s − 144.·31-s + 53.9·33-s + 107.·35-s − 150.·37-s − 52.1·39-s − 229.·41-s − 411.·43-s − 138.·45-s + 45.2·47-s + 49·49-s + 253.·51-s + 604.·53-s + 277.·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.37·5-s − 0.377·7-s + 0.333·9-s − 0.493·11-s + 0.371·13-s + 0.794·15-s − 1.20·17-s − 1.92·19-s + 0.218·21-s + 1.05·23-s + 0.895·25-s − 0.192·27-s − 1.37·29-s − 0.835·31-s + 0.284·33-s + 0.520·35-s − 0.670·37-s − 0.214·39-s − 0.874·41-s − 1.45·43-s − 0.458·45-s + 0.140·47-s + 0.142·49-s + 0.694·51-s + 1.56·53-s + 0.679·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.1587279410\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1587279410\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 + 7T \) |
good | 5 | \( 1 + 15.3T + 125T^{2} \) |
| 11 | \( 1 + 17.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 17.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 84.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 159.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 115.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 215.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 144.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 150.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 229.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 411.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 45.2T + 1.03e5T^{2} \) |
| 53 | \( 1 - 604.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 315.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 595.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 311.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 358.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 816.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 137.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 64.5T + 5.71e5T^{2} \) |
| 89 | \( 1 - 487.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.28e3T + 9.12e5T^{2} \) |
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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
−8.964963316373908878996355485549, −8.552633169797653336044227226864, −7.46274820335089144184036539590, −6.88492762520492514702679321174, −6.00865899900285077822326902680, −4.87664844951692625282179973232, −4.14243795160563601444384945171, −3.30808484270123979424960430841, −1.90744709622484546793163572686, −0.19566998075658332241712031389,
0.19566998075658332241712031389, 1.90744709622484546793163572686, 3.30808484270123979424960430841, 4.14243795160563601444384945171, 4.87664844951692625282179973232, 6.00865899900285077822326902680, 6.88492762520492514702679321174, 7.46274820335089144184036539590, 8.552633169797653336044227226864, 8.964963316373908878996355485549