| L(s) = 1 | + 27·3-s − 230.·5-s − 343·7-s + 729·9-s + 5.02e3·11-s − 1.37e4·13-s − 6.22e3·15-s + 3.24e4·17-s − 9.65e3·19-s − 9.26e3·21-s − 3.20e4·23-s − 2.50e4·25-s + 1.96e4·27-s + 1.03e5·29-s − 2.41e5·31-s + 1.35e5·33-s + 7.90e4·35-s − 1.27e5·37-s − 3.71e5·39-s + 6.07e5·41-s − 4.43e5·43-s − 1.67e5·45-s − 6.91e5·47-s + 1.17e5·49-s + 8.76e5·51-s + 6.72e5·53-s − 1.15e6·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.824·5-s − 0.377·7-s + 0.333·9-s + 1.13·11-s − 1.73·13-s − 0.475·15-s + 1.60·17-s − 0.322·19-s − 0.218·21-s − 0.548·23-s − 0.320·25-s + 0.192·27-s + 0.789·29-s − 1.45·31-s + 0.657·33-s + 0.311·35-s − 0.412·37-s − 1.00·39-s + 1.37·41-s − 0.851·43-s − 0.274·45-s − 0.971·47-s + 0.142·49-s + 0.925·51-s + 0.620·53-s − 0.938·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.927949632\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.927949632\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 27T \) |
| 7 | \( 1 + 343T \) |
| good | 5 | \( 1 + 230.T + 7.81e4T^{2} \) |
| 11 | \( 1 - 5.02e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.37e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 3.24e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 9.65e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 3.20e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.03e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.41e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.27e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 6.07e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 4.43e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 6.91e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 6.72e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.58e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.53e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 4.20e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.51e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 5.47e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 6.75e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 8.36e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 5.17e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.06e7T + 8.07e13T^{2} \) |
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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
−10.02652879306145486442364440674, −9.562296904638144990705557927708, −8.379637625561333768721237188755, −7.54926331324622226392130505201, −6.78875199521879339120517070253, −5.37123659649004823172319188562, −4.10948593274652829980446088802, −3.36042625926871608504623968935, −2.08580767322335124267361858039, −0.63422899983295184512839642094,
0.63422899983295184512839642094, 2.08580767322335124267361858039, 3.36042625926871608504623968935, 4.10948593274652829980446088802, 5.37123659649004823172319188562, 6.78875199521879339120517070253, 7.54926331324622226392130505201, 8.379637625561333768721237188755, 9.562296904638144990705557927708, 10.02652879306145486442364440674
