| L(s) = 1 | + 1.48i·3-s + 6.36·5-s + 0.636i·7-s + 6.78·9-s − 19.6i·11-s + 4.35·13-s + 9.47i·15-s + 5.07·17-s − 3.42i·19-s − 0.947·21-s − 23.9i·23-s + 15.4·25-s + 23.5i·27-s − 5.38·29-s + 32.7i·31-s + ⋯ |
| L(s) = 1 | + 0.496i·3-s + 1.27·5-s + 0.0908i·7-s + 0.753·9-s − 1.78i·11-s + 0.334·13-s + 0.631i·15-s + 0.298·17-s − 0.180i·19-s − 0.0451·21-s − 1.03i·23-s + 0.619·25-s + 0.870i·27-s − 0.185·29-s + 1.05i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.352149951\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.352149951\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 + 5.38T \) |
| good | 3 | \( 1 - 1.48iT - 9T^{2} \) |
| 5 | \( 1 - 6.36T + 25T^{2} \) |
| 7 | \( 1 - 0.636iT - 49T^{2} \) |
| 11 | \( 1 + 19.6iT - 121T^{2} \) |
| 13 | \( 1 - 4.35T + 169T^{2} \) |
| 17 | \( 1 - 5.07T + 289T^{2} \) |
| 19 | \( 1 + 3.42iT - 361T^{2} \) |
| 23 | \( 1 + 23.9iT - 529T^{2} \) |
| 31 | \( 1 - 32.7iT - 961T^{2} \) |
| 37 | \( 1 - 18.0T + 1.36e3T^{2} \) |
| 41 | \( 1 + 14.5T + 1.68e3T^{2} \) |
| 43 | \( 1 - 14.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 39.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 28.1T + 2.80e3T^{2} \) |
| 59 | \( 1 - 74.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 33.3T + 3.72e3T^{2} \) |
| 67 | \( 1 - 74.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 62.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 2.50T + 5.32e3T^{2} \) |
| 79 | \( 1 - 99.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 157. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 106.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 118.T + 9.40e3T^{2} \) |
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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
−10.56553767896360384195963885732, −10.09440658525321784038915884935, −9.048998255554475579139938208560, −8.436801211183576851886873287420, −6.95342746553301462664839081926, −5.99275121146390273832298618088, −5.28380708836737914043125200156, −3.94346243542837705611811461971, −2.71590894028002397696095506733, −1.16787883547691611770398568899,
1.46852962196002297840704919923, 2.24880722251675998041413570292, 4.05671854423487155910783824489, 5.21264743275753330340697407591, 6.24036232359415962574997529982, 7.12813088647977597671339434629, 7.88030510652892183645906045169, 9.482342282153325579082024543668, 9.708168391571872580931980770136, 10.62671669070903426191140702443
