| L(s) = 1 | − 1.73i·3-s + 2.97i·5-s + (−3.90 + 5.80i)7-s − 2.99·9-s − 5.79·11-s − 18.9i·13-s + 5.14·15-s − 18.0i·17-s − 0.0874i·19-s + (10.0 + 6.76i)21-s + 42.6·23-s + 16.1·25-s + 5.19i·27-s + 17.9·29-s + 38.0i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.594i·5-s + (−0.558 + 0.829i)7-s − 0.333·9-s − 0.527·11-s − 1.45i·13-s + 0.343·15-s − 1.06i·17-s − 0.00460i·19-s + (0.479 + 0.322i)21-s + 1.85·23-s + 0.646·25-s + 0.192i·27-s + 0.619·29-s + 1.22i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.829 + 0.558i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.829 + 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.586537403\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.586537403\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| 7 | \( 1 + (3.90 - 5.80i)T \) |
| good | 5 | \( 1 - 2.97iT - 25T^{2} \) |
| 11 | \( 1 + 5.79T + 121T^{2} \) |
| 13 | \( 1 + 18.9iT - 169T^{2} \) |
| 17 | \( 1 + 18.0iT - 289T^{2} \) |
| 19 | \( 1 + 0.0874iT - 361T^{2} \) |
| 23 | \( 1 - 42.6T + 529T^{2} \) |
| 29 | \( 1 - 17.9T + 841T^{2} \) |
| 31 | \( 1 - 38.0iT - 961T^{2} \) |
| 37 | \( 1 - 68.7T + 1.36e3T^{2} \) |
| 41 | \( 1 - 3.58iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 50.5T + 1.84e3T^{2} \) |
| 47 | \( 1 + 19.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 61.5T + 2.80e3T^{2} \) |
| 59 | \( 1 + 10.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 79.8iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 52.9T + 4.48e3T^{2} \) |
| 71 | \( 1 + 102.T + 5.04e3T^{2} \) |
| 73 | \( 1 - 104. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 25.0T + 6.24e3T^{2} \) |
| 83 | \( 1 + 84.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 29.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 8.26iT - 9.40e3T^{2} \) |
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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
−10.32637888209767407583743118040, −9.288403673101187913240307997967, −8.473836854306582386538337241584, −7.46417154478340901843444631941, −6.76738542130010723380848376869, −5.74606695083816164036702355604, −4.96956277111859379729956337614, −2.92997290448861114274912895777, −2.81891771732194271608955191863, −0.75151984669332498782456963310,
0.999646797080205455257704479619, 2.72941157588963025105952438461, 4.08617521919709340498410865462, 4.60057214979665674316103024609, 5.87063919074858865209780992040, 6.82457608179535702664996505850, 7.79197399486092451951049704011, 8.936387303300351169873557288348, 9.392923253241025462409975414637, 10.44482920200117614124009396185
