L(s) = 1 | + (1 − 2.82i)3-s + 5.65i·5-s − 6·7-s + (−7.00 − 5.65i)9-s + 5.65i·11-s + 10·13-s + (16.0 + 5.65i)15-s − 22.6i·17-s + 2·19-s + (−6 + 16.9i)21-s − 11.3i·23-s − 7.00·25-s + (−23.0 + 14.1i)27-s + 16.9i·29-s − 22·31-s + ⋯ |
L(s) = 1 | + (0.333 − 0.942i)3-s + 1.13i·5-s − 0.857·7-s + (−0.777 − 0.628i)9-s + 0.514i·11-s + 0.769·13-s + (1.06 + 0.377i)15-s − 1.33i·17-s + 0.105·19-s + (−0.285 + 0.808i)21-s − 0.491i·23-s − 0.280·25-s + (−0.851 + 0.523i)27-s + 0.585i·29-s − 0.709·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.333i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.942 + 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.928910 - 0.159375i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.928910 - 0.159375i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1 + 2.82i)T \) |
good | 5 | \( 1 - 5.65iT - 25T^{2} \) |
| 7 | \( 1 + 6T + 49T^{2} \) |
| 11 | \( 1 - 5.65iT - 121T^{2} \) |
| 13 | \( 1 - 10T + 169T^{2} \) |
| 17 | \( 1 + 22.6iT - 289T^{2} \) |
| 19 | \( 1 - 2T + 361T^{2} \) |
| 23 | \( 1 + 11.3iT - 529T^{2} \) |
| 29 | \( 1 - 16.9iT - 841T^{2} \) |
| 31 | \( 1 + 22T + 961T^{2} \) |
| 37 | \( 1 + 6T + 1.36e3T^{2} \) |
| 41 | \( 1 - 33.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 82T + 1.84e3T^{2} \) |
| 47 | \( 1 - 67.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 62.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 73.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 86T + 3.72e3T^{2} \) |
| 67 | \( 1 - 2T + 4.48e3T^{2} \) |
| 71 | \( 1 + 124. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 82T + 5.32e3T^{2} \) |
| 79 | \( 1 - 10T + 6.24e3T^{2} \) |
| 83 | \( 1 + 73.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 33.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 94T + 9.40e3T^{2} \) |
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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
−17.93329962250395857674990055759, −16.16509422432547218507615041236, −14.73168390092593549178382905593, −13.70006705163429983926489973324, −12.46828269382948081457220826153, −10.99342490422542368991926475278, −9.311841927145405240809802494724, −7.40713148646346123493409827011, −6.36573606719747848689709829916, −2.98246483718788002904300333538,
3.84755496424067939340094305126, 5.72798389202971793650888671793, 8.396629698352459685727855168211, 9.396373175556693518577016958289, 10.84205192665977700251879871973, 12.61782911242619031988447152576, 13.77100009717908332975017242392, 15.41817941221427843923117773211, 16.28425281770056576603944690490, 17.14668355749950927793260670781