L(s) = 1 | + (2.22 + 1.28i)3-s + 3.56i·7-s + (1.79 + 3.11i)9-s + 4.56·11-s + (0.866 − 0.5i)13-s + (−4.96 − 2.86i)17-s + (4.35 − 0.221i)19-s + (−4.58 + 7.93i)21-s + (4.84 − 2.79i)23-s + 1.53i·27-s + (3.38 + 5.86i)29-s + 4.59·31-s + (10.1 + 5.86i)33-s + 3.56i·37-s + 2.56·39-s + ⋯ |
L(s) = 1 | + (1.28 + 0.741i)3-s + 1.34i·7-s + (0.599 + 1.03i)9-s + 1.37·11-s + (0.240 − 0.138i)13-s + (−1.20 − 0.695i)17-s + (0.998 − 0.0507i)19-s + (−1.00 + 1.73i)21-s + (1.01 − 0.583i)23-s + 0.296i·27-s + (0.628 + 1.08i)29-s + 0.826·31-s + (1.76 + 1.02i)33-s + 0.586i·37-s + 0.411·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.198 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.011104464\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.011104464\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-4.35 + 0.221i)T \) |
good | 3 | \( 1 + (-2.22 - 1.28i)T + (1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 - 3.56iT - 7T^{2} \) |
| 11 | \( 1 - 4.56T + 11T^{2} \) |
| 13 | \( 1 + (-0.866 + 0.5i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (4.96 + 2.86i)T + (8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-4.84 + 2.79i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.38 - 5.86i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 4.59T + 31T^{2} \) |
| 37 | \( 1 - 3.56iT - 37T^{2} \) |
| 41 | \( 1 + (5.35 - 9.27i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (9.79 + 5.65i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (5.86 - 3.38i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.94 + 1.70i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3.38 - 5.86i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.06 + 7.04i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.48 - 3.16i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.515 + 0.892i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (11.0 + 6.36i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.43 + 4.21i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 9.40iT - 83T^{2} \) |
| 89 | \( 1 + (-6.13 - 10.6i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.27 + 1.31i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.309513922050384408746175654251, −8.641917040610491878845461199024, −8.341324749972574596026141723251, −6.97193715897988098242182811561, −6.34221817077805003970009513637, −5.06142727727122008972986291005, −4.47442856083546650546176743773, −3.20375408548605782681963070250, −2.86234447788486866663245201231, −1.58689258098948048546239893077,
1.04956958593423298981825162783, 1.86272089282453797816172552924, 3.18944196904042129440665691987, 3.83473836477659851878983711562, 4.66735123076153607749901671052, 6.23598839867054229115330276982, 6.93642892461039073201887242957, 7.39137326157231773984834684588, 8.319092326215861603820479631276, 8.882408196427565187886438379808