L(s) = 1 | + (1.43 + 0.830i)3-s + (−7.27 + 4.20i)5-s + (3.99 + 5.74i)7-s + (−3.12 − 5.40i)9-s + (2.60 − 4.51i)11-s + 4.88i·13-s − 13.9·15-s + (6.68 + 3.86i)17-s + (−30.6 + 17.6i)19-s + (0.979 + 11.5i)21-s + (−13.3 − 23.0i)23-s + (22.7 − 39.4i)25-s − 25.3i·27-s − 45.1·29-s + (−35.0 − 20.2i)31-s + ⋯ |
L(s) = 1 | + (0.479 + 0.276i)3-s + (−1.45 + 0.840i)5-s + (0.571 + 0.820i)7-s + (−0.346 − 0.600i)9-s + (0.237 − 0.410i)11-s + 0.375i·13-s − 0.930·15-s + (0.393 + 0.227i)17-s + (−1.61 + 0.929i)19-s + (0.0466 + 0.551i)21-s + (−0.578 − 1.00i)23-s + (0.911 − 1.57i)25-s − 0.937i·27-s − 1.55·29-s + (−1.12 − 0.652i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.209i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.977 + 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4010558648\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4010558648\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-3.99 - 5.74i)T \) |
good | 3 | \( 1 + (-1.43 - 0.830i)T + (4.5 + 7.79i)T^{2} \) |
| 5 | \( 1 + (7.27 - 4.20i)T + (12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (-2.60 + 4.51i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 - 4.88iT - 169T^{2} \) |
| 17 | \( 1 + (-6.68 - 3.86i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (30.6 - 17.6i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (13.3 + 23.0i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + 45.1T + 841T^{2} \) |
| 31 | \( 1 + (35.0 + 20.2i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-3.97 - 6.89i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 - 26.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 0.403T + 1.84e3T^{2} \) |
| 47 | \( 1 + (28.2 - 16.3i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-40.6 + 70.3i)T + (-1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-7.62 - 4.40i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-25.3 + 14.6i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (43.7 - 75.8i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 27.5T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-75.3 - 43.5i)T + (2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-60.8 - 105. i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + 46.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (52.7 - 30.4i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 - 66.4iT - 9.40e3T^{2} \) |
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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
−11.40439211287664085855477054807, −10.67536931605912985148555819523, −9.438699404354485210472707934418, −8.364742242612701127578754857304, −8.083205403426607611170276751192, −6.76400757495414391123105770978, −5.82601396425609940560904577924, −4.16029468896077516862604947318, −3.59830164462801805134329085710, −2.29531796915920506570507593898,
0.15204883821650278857141968399, 1.78700237856870339438443010453, 3.56043121212941294178191273802, 4.40291168275959378608008405620, 5.35134383787069893963751886289, 7.19974319331491600221773228092, 7.66026221657067505007982959534, 8.428571313629351558308000803099, 9.201923991785489017833222210532, 10.70887988355551792693056100953