L(s) = 1 | + (0.744 + 1.56i)3-s + 0.553·5-s + (−1.89 + 2.32i)9-s − 4.65i·11-s + (−3.58 + 2.06i)13-s + (0.412 + 0.866i)15-s + (−3.62 − 6.27i)17-s + (−5.81 − 3.35i)19-s − 5.60i·23-s − 4.69·25-s + (−5.05 − 1.22i)27-s + (1.16 + 0.673i)29-s + (−0.830 − 0.479i)31-s + (7.28 − 3.47i)33-s + (3.53 − 6.12i)37-s + ⋯ |
L(s) = 1 | + (0.430 + 0.902i)3-s + 0.247·5-s + (−0.630 + 0.776i)9-s − 1.40i·11-s + (−0.993 + 0.573i)13-s + (0.106 + 0.223i)15-s + (−0.878 − 1.52i)17-s + (−1.33 − 0.770i)19-s − 1.16i·23-s − 0.938·25-s + (−0.972 − 0.234i)27-s + (0.216 + 0.125i)29-s + (−0.149 − 0.0861i)31-s + (1.26 − 0.604i)33-s + (0.581 − 1.00i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6649538766\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6649538766\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.744 - 1.56i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 0.553T + 5T^{2} \) |
| 11 | \( 1 + 4.65iT - 11T^{2} \) |
| 13 | \( 1 + (3.58 - 2.06i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (3.62 + 6.27i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (5.81 + 3.35i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 5.60iT - 23T^{2} \) |
| 29 | \( 1 + (-1.16 - 0.673i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (0.830 + 0.479i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.53 + 6.12i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.39 - 4.14i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.02 - 1.78i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.90 - 8.49i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (7.30 - 4.21i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3.89 - 6.75i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.37 - 3.10i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.68 + 2.92i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 0.407iT - 71T^{2} \) |
| 73 | \( 1 + (-7.47 + 4.31i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.318 + 0.551i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (2.78 - 4.82i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (3.46 - 6.00i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (7.48 + 4.32i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.231199815002792477885615045915, −8.495037694786262868642103882735, −7.62764151227771498925543702868, −6.58748352763764079796563503702, −5.78569676415496940389428669820, −4.68425668627073547214074353838, −4.27587751835185914337260315487, −2.87823468242832820709772617128, −2.37058685301488940392244695729, −0.20815553163180755221691102674,
1.80263247179879798009296174856, 2.18597100066109821771850863662, 3.59159230927471106882802045630, 4.51724843682111393107213823927, 5.68965094367804645793798899122, 6.43732260265795613279632447145, 7.21173927086363741735890342614, 7.939401578130154630287582864396, 8.542478441750735738314216060550, 9.593700564124261334680342645162