L(s) = 1 | + (1.44 − 0.313i)3-s + (−0.625 + 2.14i)5-s + (2.34 − 1.27i)7-s + (−0.745 + 0.340i)9-s + (4.30 + 3.72i)11-s + (−2.02 + 3.71i)13-s + (−0.229 + 3.29i)15-s + (−2.53 − 1.89i)17-s + (−0.894 + 6.22i)19-s + (2.97 − 2.58i)21-s + (−1.71 − 4.47i)23-s + (−4.21 − 2.68i)25-s + (−4.51 + 3.37i)27-s + (2.66 − 0.383i)29-s + (5.88 − 3.78i)31-s + ⋯ |
L(s) = 1 | + (0.833 − 0.181i)3-s + (−0.279 + 0.960i)5-s + (0.885 − 0.483i)7-s + (−0.248 + 0.113i)9-s + (1.29 + 1.12i)11-s + (−0.562 + 1.03i)13-s + (−0.0591 + 0.850i)15-s + (−0.613 − 0.459i)17-s + (−0.205 + 1.42i)19-s + (0.650 − 0.563i)21-s + (−0.357 − 0.934i)23-s + (−0.843 − 0.537i)25-s + (−0.868 + 0.650i)27-s + (0.495 − 0.0711i)29-s + (1.05 − 0.679i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.80799 + 1.00796i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.80799 + 1.00796i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.625 - 2.14i)T \) |
| 23 | \( 1 + (1.71 + 4.47i)T \) |
good | 3 | \( 1 + (-1.44 + 0.313i)T + (2.72 - 1.24i)T^{2} \) |
| 7 | \( 1 + (-2.34 + 1.27i)T + (3.78 - 5.88i)T^{2} \) |
| 11 | \( 1 + (-4.30 - 3.72i)T + (1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (2.02 - 3.71i)T + (-7.02 - 10.9i)T^{2} \) |
| 17 | \( 1 + (2.53 + 1.89i)T + (4.78 + 16.3i)T^{2} \) |
| 19 | \( 1 + (0.894 - 6.22i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-2.66 + 0.383i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (-5.88 + 3.78i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (-7.38 - 2.75i)T + (27.9 + 24.2i)T^{2} \) |
| 41 | \( 1 + (-2.17 + 4.75i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-0.0789 - 0.362i)T + (-39.1 + 17.8i)T^{2} \) |
| 47 | \( 1 + (-1.20 - 1.20i)T + 47iT^{2} \) |
| 53 | \( 1 + (-2.10 - 3.85i)T + (-28.6 + 44.5i)T^{2} \) |
| 59 | \( 1 + (3.63 + 12.3i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-7.08 - 11.0i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (7.83 + 0.560i)T + (66.3 + 9.53i)T^{2} \) |
| 71 | \( 1 + (-4.92 - 5.68i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-0.604 - 0.807i)T + (-20.5 + 70.0i)T^{2} \) |
| 79 | \( 1 + (-13.4 + 3.94i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (-1.57 + 4.22i)T + (-62.7 - 54.3i)T^{2} \) |
| 89 | \( 1 + (1.64 + 1.05i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (4.70 + 12.6i)T + (-73.3 + 63.5i)T^{2} \) |
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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
−10.08898390119698604394800435190, −9.420586509739012004793849964787, −8.394519260037376103819549677163, −7.70032334499870971077271970348, −6.97326793162708247714582293917, −6.20417938893876234831295260570, −4.47058376484558772644998590165, −4.05858865180410100880064355923, −2.59278868330643743502896702222, −1.79419859984344437637104611316,
0.941580001932799945534901163608, 2.45402403218806285309333320161, 3.56142065420764704987152040403, 4.55305206075056970890229591007, 5.45751021991608045576930653531, 6.43969508273424464064511828890, 7.88771600349218524719340485339, 8.353727742091459097240463358521, 8.994050970544421008926098643781, 9.531691555647967079870496660933