| L(s) = 1 | + 3.59·5-s − 9.49i·7-s − 7.23i·11-s − 9.12·13-s − 7.56·17-s + 4.35i·19-s − 15.6i·23-s − 12.0·25-s − 33.8·29-s − 6.39i·31-s − 34.1i·35-s + 47.7·37-s − 63.7·41-s + 68.2i·43-s + 30.0i·47-s + ⋯ |
| L(s) = 1 | + 0.719·5-s − 1.35i·7-s − 0.657i·11-s − 0.701·13-s − 0.444·17-s + 0.229i·19-s − 0.679i·23-s − 0.482·25-s − 1.16·29-s − 0.206i·31-s − 0.975i·35-s + 1.28·37-s − 1.55·41-s + 1.58i·43-s + 0.639i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.5i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2694459441\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2694459441\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - 4.35iT \) |
| good | 5 | \( 1 - 3.59T + 25T^{2} \) |
| 7 | \( 1 + 9.49iT - 49T^{2} \) |
| 11 | \( 1 + 7.23iT - 121T^{2} \) |
| 13 | \( 1 + 9.12T + 169T^{2} \) |
| 17 | \( 1 + 7.56T + 289T^{2} \) |
| 23 | \( 1 + 15.6iT - 529T^{2} \) |
| 29 | \( 1 + 33.8T + 841T^{2} \) |
| 31 | \( 1 + 6.39iT - 961T^{2} \) |
| 37 | \( 1 - 47.7T + 1.36e3T^{2} \) |
| 41 | \( 1 + 63.7T + 1.68e3T^{2} \) |
| 43 | \( 1 - 68.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 30.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 63.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + 48.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 0.829T + 3.72e3T^{2} \) |
| 67 | \( 1 - 65.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 5.67iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 8.07T + 5.32e3T^{2} \) |
| 79 | \( 1 - 104. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 109. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 32.2T + 7.92e3T^{2} \) |
| 97 | \( 1 + 72.7T + 9.40e3T^{2} \) |
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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
−8.042089694073365675144041978747, −7.53042985170254935037909668592, −6.59169202782254674267235456795, −6.04539780129897581776151449851, −5.01307563549503518793508096644, −4.25512347607944265573009576996, −3.36679790563192304196271389886, −2.28761401191667294142836901834, −1.21095553946277032655277571376, −0.05818415239768517490214116830,
1.83009664091759634442372914227, 2.25578792671395514923490087356, 3.33274465587929012401689344585, 4.55638146219272808670586165959, 5.36757816828104436922217704941, 5.86051134722570285053676407981, 6.78149110965808109427085271838, 7.54303493457530475782319264793, 8.457921796784278437785531058148, 9.258105917909623615856423500091
