| L(s) = 1 | + (0.415 − 0.909i)2-s + (−0.730 + 2.48i)3-s + (−0.654 − 0.755i)4-s + (−1.52 + 0.983i)5-s + (1.96 + 1.69i)6-s + (−2.20 − 1.45i)7-s + (−0.959 + 0.281i)8-s + (−3.13 − 2.01i)9-s + (0.258 + 1.80i)10-s + (−0.133 + 0.0609i)11-s + (2.35 − 1.07i)12-s + (−2.27 + 0.326i)13-s + (−2.24 + 1.40i)14-s + (−1.32 − 4.52i)15-s + (−0.142 + 0.989i)16-s + (−2.03 + 2.34i)17-s + ⋯ |
| L(s) = 1 | + (0.293 − 0.643i)2-s + (−0.421 + 1.43i)3-s + (−0.327 − 0.377i)4-s + (−0.684 + 0.439i)5-s + (0.800 + 0.693i)6-s + (−0.834 − 0.551i)7-s + (−0.339 + 0.0996i)8-s + (−1.04 − 0.672i)9-s + (0.0818 + 0.569i)10-s + (−0.0402 + 0.0183i)11-s + (0.681 − 0.311i)12-s + (−0.630 + 0.0906i)13-s + (−0.599 + 0.374i)14-s + (−0.343 − 1.16i)15-s + (−0.0355 + 0.247i)16-s + (−0.493 + 0.569i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 - 0.282i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.959 - 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0437534 + 0.302991i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0437534 + 0.302991i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.415 + 0.909i)T \) |
| 7 | \( 1 + (2.20 + 1.45i)T \) |
| 23 | \( 1 + (3.54 - 3.23i)T \) |
| good | 3 | \( 1 + (0.730 - 2.48i)T + (-2.52 - 1.62i)T^{2} \) |
| 5 | \( 1 + (1.52 - 0.983i)T + (2.07 - 4.54i)T^{2} \) |
| 11 | \( 1 + (0.133 - 0.0609i)T + (7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (2.27 - 0.326i)T + (12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (2.03 - 2.34i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (3.74 + 4.32i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (-4.85 + 5.60i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (-0.987 - 3.36i)T + (-26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (1.53 - 2.39i)T + (-15.3 - 33.6i)T^{2} \) |
| 41 | \( 1 + (-4.56 - 7.10i)T + (-17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (1.10 - 3.75i)T + (-36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + 3.45iT - 47T^{2} \) |
| 53 | \( 1 + (-11.7 - 1.69i)T + (50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (2.65 - 0.381i)T + (56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (-3.68 + 1.08i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (1.68 + 0.769i)T + (43.8 + 50.6i)T^{2} \) |
| 71 | \( 1 + (4.71 - 10.3i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (2.35 - 2.03i)T + (10.3 - 72.2i)T^{2} \) |
| 79 | \( 1 + (8.52 - 1.22i)T + (75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (13.4 + 8.63i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (1.61 + 0.473i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (10.3 - 6.67i)T + (40.2 - 88.2i)T^{2} \) |
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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
−11.67696061813394791202052134799, −11.09730988024873717348063604932, −10.18370783780799352842208285765, −9.786442972293264681048448767871, −8.611287938119957747374790685704, −7.10759104464709200447384613926, −5.94760566033031744419809541785, −4.55401383114852905867830316972, −4.00732077353759922548661163509, −2.89541960916875063973597568674,
0.19784400894982576037110682048, 2.42374128069161554063913936786, 4.15284352724288217675204859121, 5.55679344161606417451602865953, 6.42075612200879834032523708283, 7.18996866947701490013649142335, 8.109661591889666411682019408577, 8.901447056308682760783410413397, 10.30660230698546499151532593153, 11.82961704206054478027273902033
