| L(s) = 1 | + 9·3-s − 78.6·5-s + 81·9-s + 691.·11-s − 818.·13-s − 708.·15-s − 1.10e3·17-s + 573.·19-s + 2.51e3·23-s + 3.06e3·25-s + 729·27-s − 3.25e3·29-s − 1.01e4·31-s + 6.22e3·33-s + 4.86e3·37-s − 7.36e3·39-s + 1.30e4·41-s − 9.30e3·43-s − 6.37e3·45-s − 1.29e4·47-s − 9.98e3·51-s − 1.95e4·53-s − 5.44e4·55-s + 5.15e3·57-s + 2.51e4·59-s + 3.13e4·61-s + 6.44e4·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.40·5-s + 0.333·9-s + 1.72·11-s − 1.34·13-s − 0.812·15-s − 0.930·17-s + 0.364·19-s + 0.992·23-s + 0.980·25-s + 0.192·27-s − 0.719·29-s − 1.89·31-s + 0.994·33-s + 0.584·37-s − 0.775·39-s + 1.21·41-s − 0.767·43-s − 0.469·45-s − 0.852·47-s − 0.537·51-s − 0.955·53-s − 2.42·55-s + 0.210·57-s + 0.939·59-s + 1.07·61-s + 1.89·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.766590704\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.766590704\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 9T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 78.6T + 3.12e3T^{2} \) |
| 11 | \( 1 - 691.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 818.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.10e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 573.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.51e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.25e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.01e4T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.86e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.30e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 9.30e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.29e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.95e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.51e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.13e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.59e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.05e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.76e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.40e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.71e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 320.T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.12e5T + 8.58e9T^{2} \) |
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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
−9.551447390934058706702438166483, −9.134877880326552358716895645280, −8.099043227170712005772906475438, −7.26860516723582170325710347288, −6.69808121762116658713841796084, −5.03250606626888387137054384718, −4.06828331719991921553191082911, −3.42814849294869845924327700258, −2.05406851904030609848654173438, −0.62230237170248182907945090259,
0.62230237170248182907945090259, 2.05406851904030609848654173438, 3.42814849294869845924327700258, 4.06828331719991921553191082911, 5.03250606626888387137054384718, 6.69808121762116658713841796084, 7.26860516723582170325710347288, 8.099043227170712005772906475438, 9.134877880326552358716895645280, 9.551447390934058706702438166483
