| L(s) = 1 | + 1.08·3-s + 2.09i·5-s + 3.72·7-s − 7.81·9-s − 6.64i·11-s − 9.32·13-s + 2.27i·15-s − 2.02·17-s + (17.3 + 7.78i)19-s + 4.04·21-s + 7.73·23-s + 20.6·25-s − 18.2·27-s + 35.3·29-s + 37.0i·31-s + ⋯ |
| L(s) = 1 | + 0.362·3-s + 0.418i·5-s + 0.531·7-s − 0.868·9-s − 0.603i·11-s − 0.717·13-s + 0.151i·15-s − 0.118·17-s + (0.912 + 0.409i)19-s + 0.192·21-s + 0.336·23-s + 0.824·25-s − 0.677·27-s + 1.21·29-s + 1.19i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.631 - 0.775i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.631 - 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.037870721\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.037870721\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-17.3 - 7.78i)T \) |
| good | 3 | \( 1 - 1.08T + 9T^{2} \) |
| 5 | \( 1 - 2.09iT - 25T^{2} \) |
| 7 | \( 1 - 3.72T + 49T^{2} \) |
| 11 | \( 1 + 6.64iT - 121T^{2} \) |
| 13 | \( 1 + 9.32T + 169T^{2} \) |
| 17 | \( 1 + 2.02T + 289T^{2} \) |
| 23 | \( 1 - 7.73T + 529T^{2} \) |
| 29 | \( 1 - 35.3T + 841T^{2} \) |
| 31 | \( 1 - 37.0iT - 961T^{2} \) |
| 37 | \( 1 - 33.2T + 1.36e3T^{2} \) |
| 41 | \( 1 - 57.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 28.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 73.9T + 2.20e3T^{2} \) |
| 53 | \( 1 - 59.7T + 2.80e3T^{2} \) |
| 59 | \( 1 + 60.0T + 3.48e3T^{2} \) |
| 61 | \( 1 + 18.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 91.3T + 4.48e3T^{2} \) |
| 71 | \( 1 - 83.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 10.2T + 5.32e3T^{2} \) |
| 79 | \( 1 - 58.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 5.17iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 145. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 39.3iT - 9.40e3T^{2} \) |
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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
−9.597621045108090237219425614128, −8.735414029423260772826887840461, −8.109755902247530647502669471088, −7.29163373734544657400093906336, −6.33461982252486214866199663844, −5.40992438327716469030138678374, −4.55312766595059084185157182690, −3.17595800384663771713334887178, −2.65708261902278791296833467150, −1.09173636590174192772264625144,
0.66588755598833397691436009377, 2.15442358516925235278216766232, 3.02322098723613741937124875354, 4.37015480572428130020217925930, 5.07151401826554187858434331169, 5.95703588589237221621227929016, 7.19544147113249679601238748095, 7.76157396445846026610462116509, 8.766351393955943878906443799806, 9.198916424208866911610427177256
