L(s) = 1 | + (−0.876 − 1.51i)5-s − i·7-s + 1.75i·11-s + (−1.5 − 0.866i)13-s + (2.39 + 4.14i)17-s + (1.73 + 4i)19-s + (−5.66 − 3.27i)23-s + (0.964 − 1.66i)25-s + (7.18 + 4.14i)29-s + 7.73·31-s + (−1.51 + 0.876i)35-s − 7.73i·37-s + (−0.401 + 0.232i)43-s + (8.29 + 4.78i)47-s + 6·49-s + ⋯ |
L(s) = 1 | + (−0.391 − 0.678i)5-s − 0.377i·7-s + 0.528i·11-s + (−0.416 − 0.240i)13-s + (0.580 + 1.00i)17-s + (0.397 + 0.917i)19-s + (−1.18 − 0.681i)23-s + (0.192 − 0.333i)25-s + (1.33 + 0.770i)29-s + 1.38·31-s + (−0.256 + 0.148i)35-s − 1.27i·37-s + (−0.0612 + 0.0353i)43-s + (1.20 + 0.698i)47-s + 0.857·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.740 + 0.671i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.740 + 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.594355400\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.594355400\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-1.73 - 4i)T \) |
good | 5 | \( 1 + (0.876 + 1.51i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + iT - 7T^{2} \) |
| 11 | \( 1 - 1.75iT - 11T^{2} \) |
| 13 | \( 1 + (1.5 + 0.866i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.39 - 4.14i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (5.66 + 3.27i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-7.18 - 4.14i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 7.73T + 31T^{2} \) |
| 37 | \( 1 + 7.73iT - 37T^{2} \) |
| 41 | \( 1 + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.401 - 0.232i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-8.29 - 4.78i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (9.81 + 5.66i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.66 - 9.81i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.96 + 8.59i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.401 + 0.696i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (7.18 + 12.4i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (7.69 + 13.3i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.33 - 2.30i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 1.28iT - 83T^{2} \) |
| 89 | \( 1 + (4.55 + 2.62i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.19 + 3i)T + (48.5 - 84.0i)T^{2} \) |
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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
−8.525744576290693054942339957663, −8.064236622433855237245767541431, −7.38781160715705122175874400904, −6.39313337800219385079863826826, −5.66850621688628792815531395783, −4.61168427249569803929769177383, −4.16154745205038327112170740645, −3.08842471403192934616034397748, −1.85632210600389058134530562787, −0.69742773693096301001568402172,
0.911026782067512745171915955972, 2.55116620087242608614297425554, 3.03405548518104319931332782983, 4.14137377678902367928291906087, 5.03431663466865645536997194747, 5.86567884054825367124523665019, 6.73521642190318580115562495115, 7.35012625497147402396565581426, 8.135742181519414022124622543727, 8.839943662834866357868267202797