| L(s) = 1 | − 1.41i·2-s − 2.00·4-s + 6.06i·5-s + 2.82i·8-s + 8.58·10-s − 12.1i·11-s + 18.5·13-s + 4.00·16-s + 10.9i·17-s − 20·19-s − 12.1i·20-s − 17.1·22-s − 12.1i·23-s − 11.8·25-s − 26.2i·26-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s + 1.21i·5-s + 0.353i·8-s + 0.858·10-s − 1.10i·11-s + 1.42·13-s + 0.250·16-s + 0.641i·17-s − 1.05·19-s − 0.606i·20-s − 0.780·22-s − 0.527i·23-s − 0.473·25-s − 1.01i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.597385419\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.597385419\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 6.06iT - 25T^{2} \) |
| 11 | \( 1 + 12.1iT - 121T^{2} \) |
| 13 | \( 1 - 18.5T + 169T^{2} \) |
| 17 | \( 1 - 10.9iT - 289T^{2} \) |
| 19 | \( 1 + 20T + 361T^{2} \) |
| 23 | \( 1 + 12.1iT - 529T^{2} \) |
| 29 | \( 1 - 41.8iT - 841T^{2} \) |
| 31 | \( 1 + 25.1T + 961T^{2} \) |
| 37 | \( 1 - 38T + 1.36e3T^{2} \) |
| 41 | \( 1 - 60.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 83.4T + 1.84e3T^{2} \) |
| 47 | \( 1 - 16.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 94.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 58.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 15.6T + 3.72e3T^{2} \) |
| 67 | \( 1 + 132.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 12.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 76.9T + 5.32e3T^{2} \) |
| 79 | \( 1 - 33.6T + 6.24e3T^{2} \) |
| 83 | \( 1 - 60.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 4.77iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 188.T + 9.40e3T^{2} \) |
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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
−10.48843090666776772203703875981, −9.132636708222898513875041253091, −8.553397975171837808105081769015, −7.56192047935155680089750270631, −6.31090114922020771723968919292, −5.93796801670953561996555737755, −4.31819383622217424229097272028, −3.43204930669500127619164168794, −2.65499430239307134839998459015, −1.20517561832429915827037626936,
0.60158639314093212695543247492, 1.98163545217554938788899857921, 3.87090155293225851281059968014, 4.55910238973426438417750038752, 5.50859034353940230781116776037, 6.32478025212860335265152778726, 7.39476512965285040014171472071, 8.174883950822671422335241177150, 8.996836718513461131998685681454, 9.484233711258791741243393271729
