L(s) = 1 | − 1.73·3-s + 1.36i·5-s + 1.24i·7-s + 2.99·9-s + 5.79·11-s − 16.3i·13-s − 2.36i·15-s − 5.01·17-s − 26.1·19-s − 2.15i·21-s + 25.1i·23-s + 23.1·25-s − 5.19·27-s − 32.7i·29-s + 1.01i·31-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.272i·5-s + 0.177i·7-s + 0.333·9-s + 0.527·11-s − 1.26i·13-s − 0.157i·15-s − 0.294·17-s − 1.37·19-s − 0.102i·21-s + 1.09i·23-s + 0.925·25-s − 0.192·27-s − 1.13i·29-s + 0.0328i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.319436509\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.319436509\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73T \) |
good | 5 | \( 1 - 1.36iT - 25T^{2} \) |
| 7 | \( 1 - 1.24iT - 49T^{2} \) |
| 11 | \( 1 - 5.79T + 121T^{2} \) |
| 13 | \( 1 + 16.3iT - 169T^{2} \) |
| 17 | \( 1 + 5.01T + 289T^{2} \) |
| 19 | \( 1 + 26.1T + 361T^{2} \) |
| 23 | \( 1 - 25.1iT - 529T^{2} \) |
| 29 | \( 1 + 32.7iT - 841T^{2} \) |
| 31 | \( 1 - 1.01iT - 961T^{2} \) |
| 37 | \( 1 - 14.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 72.5T + 1.68e3T^{2} \) |
| 43 | \( 1 - 33.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + 66.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 54.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 20.5T + 3.48e3T^{2} \) |
| 61 | \( 1 + 111. iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 60.9T + 4.48e3T^{2} \) |
| 71 | \( 1 + 80.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 30.0T + 5.32e3T^{2} \) |
| 79 | \( 1 - 80.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 113.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 21.0T + 7.92e3T^{2} \) |
| 97 | \( 1 - 160.T + 9.40e3T^{2} \) |
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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
−10.17681368135124298510454381426, −9.223726955826163709675518706349, −8.274846918674546736148317406263, −7.35711392018329706928919370789, −6.38055759908235508086329038232, −5.69285233303383193040675045775, −4.61707947118745667438303586063, −3.53247301067311527292422251202, −2.21397156227695742506374617618, −0.59858139258565110675344726549,
1.05518106369720454240977413209, 2.42314365605901555658170415324, 4.15748227099426401956943071386, 4.58331357289673242200380778191, 5.96267452475235089369516363651, 6.63961050675239495045112083226, 7.46358899236528323486211609809, 8.890662472287664369426868627618, 9.082210917483977751730114802764, 10.53577513427936346034740868240