| L(s) = 1 | + 0.644i·3-s + 2.32·5-s − 8.62i·7-s + 8.58·9-s + 19.2i·11-s − 13.8·13-s + 1.50i·15-s − 12.5·17-s + 4.35i·19-s + 5.56·21-s + 37.4i·23-s − 19.5·25-s + 11.3i·27-s + 6.36·29-s + 5.44i·31-s + ⋯ |
| L(s) = 1 | + 0.214i·3-s + 0.465·5-s − 1.23i·7-s + 0.953·9-s + 1.75i·11-s − 1.06·13-s + 0.100i·15-s − 0.736·17-s + 0.229i·19-s + 0.264·21-s + 1.63i·23-s − 0.783·25-s + 0.419i·27-s + 0.219·29-s + 0.175i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.629498105\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.629498105\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - 4.35iT \) |
| good | 3 | \( 1 - 0.644iT - 9T^{2} \) |
| 5 | \( 1 - 2.32T + 25T^{2} \) |
| 7 | \( 1 + 8.62iT - 49T^{2} \) |
| 11 | \( 1 - 19.2iT - 121T^{2} \) |
| 13 | \( 1 + 13.8T + 169T^{2} \) |
| 17 | \( 1 + 12.5T + 289T^{2} \) |
| 23 | \( 1 - 37.4iT - 529T^{2} \) |
| 29 | \( 1 - 6.36T + 841T^{2} \) |
| 31 | \( 1 - 5.44iT - 961T^{2} \) |
| 37 | \( 1 + 20.9T + 1.36e3T^{2} \) |
| 41 | \( 1 - 72.8T + 1.68e3T^{2} \) |
| 43 | \( 1 - 10.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 32.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 42.9T + 2.80e3T^{2} \) |
| 59 | \( 1 - 38.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 25.5T + 3.72e3T^{2} \) |
| 67 | \( 1 + 65.3iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 18.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 72.8T + 5.32e3T^{2} \) |
| 79 | \( 1 - 139. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 94.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 33.3T + 7.92e3T^{2} \) |
| 97 | \( 1 + 150.T + 9.40e3T^{2} \) |
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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
−9.678496939189310730916886865348, −9.438731459144724587210358992697, −7.74146853589994951845301521211, −7.32107323314539222495559415044, −6.70025425924230565445555443643, −5.34150077730807148014196242003, −4.44791488300972621347974333733, −3.93539152771761797331138400818, −2.30529825390530564985715418198, −1.35530030499754168707078404638,
0.47624022070075780073218194372, 2.08122554193993352562248688572, 2.77984764761750511922218603479, 4.17070722253972711414309927287, 5.21423775126910088890165987948, 6.02146250502259195006646181847, 6.68206586802564243593480145986, 7.77449834358508038097303049869, 8.674249175779029172257660194231, 9.182477719205499727107056433956
