| L(s) = 1 | + (−1.75 + 0.951i)2-s + (2.19 − 3.34i)4-s + 1.86·5-s − 11.3i·7-s + (−0.672 + 7.97i)8-s + (−3.27 + 1.77i)10-s − 5.87i·11-s − 13.9·13-s + (10.7 + 19.9i)14-s + (−6.39 − 14.6i)16-s − 11.0·17-s + 9.34i·19-s + (4.08 − 6.23i)20-s + (5.58 + 10.3i)22-s + 30.4i·23-s + ⋯ |
| L(s) = 1 | + (−0.879 + 0.475i)2-s + (0.547 − 0.836i)4-s + 0.372·5-s − 1.61i·7-s + (−0.0840 + 0.996i)8-s + (−0.327 + 0.177i)10-s − 0.533i·11-s − 1.07·13-s + (0.768 + 1.42i)14-s + (−0.399 − 0.916i)16-s − 0.651·17-s + 0.491i·19-s + (0.204 − 0.311i)20-s + (0.253 + 0.469i)22-s + 1.32i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.547 + 0.836i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.547 + 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.266629 - 0.493259i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.266629 - 0.493259i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.75 - 0.951i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 1.86T + 25T^{2} \) |
| 7 | \( 1 + 11.3iT - 49T^{2} \) |
| 11 | \( 1 + 5.87iT - 121T^{2} \) |
| 13 | \( 1 + 13.9T + 169T^{2} \) |
| 17 | \( 1 + 11.0T + 289T^{2} \) |
| 19 | \( 1 - 9.34iT - 361T^{2} \) |
| 23 | \( 1 - 30.4iT - 529T^{2} \) |
| 29 | \( 1 + 45.5T + 841T^{2} \) |
| 31 | \( 1 + 49.5iT - 961T^{2} \) |
| 37 | \( 1 - 48.9T + 1.36e3T^{2} \) |
| 41 | \( 1 + 14.8T + 1.68e3T^{2} \) |
| 43 | \( 1 - 5.02iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 83.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 53.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + 98.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 20.4T + 3.72e3T^{2} \) |
| 67 | \( 1 - 17.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 52.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 98.1T + 5.32e3T^{2} \) |
| 79 | \( 1 + 3.04iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 101. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 17.0T + 7.92e3T^{2} \) |
| 97 | \( 1 + 52.0T + 9.40e3T^{2} \) |
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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
−10.94968329264046178591445382318, −9.824280821157405252839843939859, −9.525154730901921000888313431703, −7.945455202210446214518830502309, −7.45595270104995329475635362788, −6.41463769323066282772977344546, −5.30736832316189191765983893085, −3.85706678971392270971216109182, −1.93680067336456159358142846598, −0.32708483547886811856257630468,
2.01523218612997633520314688167, 2.77166809019414251983167633317, 4.64543502249655902662061803380, 5.97151258498658838714630486341, 7.05752672669363682068483504439, 8.183415524955897942580848246430, 9.163861386165255313695121602393, 9.589288043616126533118025612231, 10.73367044432997071754465214587, 11.69343155099150540060037288534
