| L(s) = 1 | + 1.59e6·3-s − 4.07e9·5-s + 1.89e11·7-s + 2.54e12·9-s + 1.71e14·11-s − 1.80e15·13-s − 6.50e15·15-s − 7.39e16·17-s − 5.50e16·19-s + 3.02e17·21-s + 1.48e18·23-s + 9.17e18·25-s + 4.05e18·27-s + 6.45e19·29-s − 6.82e19·31-s + 2.72e20·33-s − 7.73e20·35-s + 5.98e20·37-s − 2.87e21·39-s − 7.95e20·41-s + 8.17e21·43-s − 1.03e22·45-s + 2.61e22·47-s − 2.97e22·49-s − 1.17e23·51-s − 2.61e23·53-s − 6.98e23·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.49·5-s + 0.739·7-s + 0.333·9-s + 1.49·11-s − 1.65·13-s − 0.862·15-s − 1.81·17-s − 0.300·19-s + 0.426·21-s + 0.612·23-s + 1.23·25-s + 0.192·27-s + 1.16·29-s − 0.502·31-s + 0.863·33-s − 1.10·35-s + 0.403·37-s − 0.952·39-s − 0.134·41-s + 0.725·43-s − 0.497·45-s + 0.698·47-s − 0.453·49-s − 1.04·51-s − 1.37·53-s − 2.23·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(28-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+27/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(14)\) |
\(\approx\) |
\(1.742096664\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.742096664\) |
| \(L(\frac{29}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.59e6T \) |
| good | 5 | \( 1 + 4.07e9T + 7.45e18T^{2} \) |
| 7 | \( 1 - 1.89e11T + 6.57e22T^{2} \) |
| 11 | \( 1 - 1.71e14T + 1.31e28T^{2} \) |
| 13 | \( 1 + 1.80e15T + 1.19e30T^{2} \) |
| 17 | \( 1 + 7.39e16T + 1.66e33T^{2} \) |
| 19 | \( 1 + 5.50e16T + 3.36e34T^{2} \) |
| 23 | \( 1 - 1.48e18T + 5.84e36T^{2} \) |
| 29 | \( 1 - 6.45e19T + 3.05e39T^{2} \) |
| 31 | \( 1 + 6.82e19T + 1.84e40T^{2} \) |
| 37 | \( 1 - 5.98e20T + 2.19e42T^{2} \) |
| 41 | \( 1 + 7.95e20T + 3.50e43T^{2} \) |
| 43 | \( 1 - 8.17e21T + 1.26e44T^{2} \) |
| 47 | \( 1 - 2.61e22T + 1.40e45T^{2} \) |
| 53 | \( 1 + 2.61e23T + 3.59e46T^{2} \) |
| 59 | \( 1 + 3.67e23T + 6.50e47T^{2} \) |
| 61 | \( 1 - 9.65e22T + 1.59e48T^{2} \) |
| 67 | \( 1 + 8.38e24T + 2.01e49T^{2} \) |
| 71 | \( 1 - 1.34e25T + 9.63e49T^{2} \) |
| 73 | \( 1 + 1.30e25T + 2.04e50T^{2} \) |
| 79 | \( 1 - 2.90e25T + 1.72e51T^{2} \) |
| 83 | \( 1 + 5.64e24T + 6.53e51T^{2} \) |
| 89 | \( 1 + 3.97e26T + 4.30e52T^{2} \) |
| 97 | \( 1 - 2.01e26T + 4.39e53T^{2} \) |
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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
−10.94436693500532443005117157072, −9.330698331600928852650419157591, −8.495997507737936948496271897935, −7.48621736765211734997499098291, −6.70625691975758057482531836782, −4.59647152603746605198860374674, −4.28148016975867483145426968632, −2.96148040917870694438810242911, −1.81038388398900894849820668871, −0.52071941934844237508838271620,
0.52071941934844237508838271620, 1.81038388398900894849820668871, 2.96148040917870694438810242911, 4.28148016975867483145426968632, 4.59647152603746605198860374674, 6.70625691975758057482531836782, 7.48621736765211734997499098291, 8.495997507737936948496271897935, 9.330698331600928852650419157591, 10.94436693500532443005117157072
