L(s) = 1 | + (1.07 + 1.68i)2-s + (−1.70 + 3.61i)4-s + 6.66·5-s + 12.1i·7-s + (−7.93 + 0.984i)8-s + (7.13 + 11.2i)10-s + 4.93i·11-s + 11.2·13-s + (−20.5 + 13.0i)14-s + (−10.1 − 12.3i)16-s − 17.2·17-s + 4.35i·19-s + (−11.3 + 24.1i)20-s + (−8.34 + 5.28i)22-s − 16.0i·23-s + ⋯ |
L(s) = 1 | + (0.535 + 0.844i)2-s + (−0.427 + 0.904i)4-s + 1.33·5-s + 1.73i·7-s + (−0.992 + 0.123i)8-s + (0.713 + 1.12i)10-s + 0.448i·11-s + 0.865·13-s + (−1.46 + 0.928i)14-s + (−0.635 − 0.772i)16-s − 1.01·17-s + 0.229i·19-s + (−0.569 + 1.20i)20-s + (−0.379 + 0.240i)22-s − 0.697i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 - 0.427i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.904 - 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.699288265\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.699288265\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.07 - 1.68i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 - 4.35iT \) |
good | 5 | \( 1 - 6.66T + 25T^{2} \) |
| 7 | \( 1 - 12.1iT - 49T^{2} \) |
| 11 | \( 1 - 4.93iT - 121T^{2} \) |
| 13 | \( 1 - 11.2T + 169T^{2} \) |
| 17 | \( 1 + 17.2T + 289T^{2} \) |
| 23 | \( 1 + 16.0iT - 529T^{2} \) |
| 29 | \( 1 - 50.3T + 841T^{2} \) |
| 31 | \( 1 + 6.35iT - 961T^{2} \) |
| 37 | \( 1 + 36.0T + 1.36e3T^{2} \) |
| 41 | \( 1 + 11.8T + 1.68e3T^{2} \) |
| 43 | \( 1 - 59.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 35.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 38.6T + 2.80e3T^{2} \) |
| 59 | \( 1 - 89.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 85.8T + 3.72e3T^{2} \) |
| 67 | \( 1 + 103. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 67.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 109.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 66.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 4.01iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 53.9T + 7.92e3T^{2} \) |
| 97 | \( 1 + 62.8T + 9.40e3T^{2} \) |
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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
−10.54120184195534026444564124112, −9.440084633226229460340648491620, −8.843926560340620117446239214038, −8.221136435807052558005813587160, −6.64662657074260075717349753300, −6.19372849165020165303293892632, −5.42730972713554265243547844377, −4.56469087173789182345337734039, −2.93274895910463685705994295160, −2.03412175472083696808100226163,
0.812579165409079702924766855990, 1.84463214853437319558015833482, 3.23130240498919707654507548369, 4.20268305688932180783117805512, 5.19239680543573254758756732579, 6.26373168942213432502980719089, 6.89729185441172324326862433920, 8.446530835803055144797835075574, 9.346678114988398840866540772394, 10.23332449248106680595883477186