L(s) = 1 | + (1.30 + 2.26i)3-s + (0.0495 − 0.0858i)5-s + (−0.841 + 1.45i)7-s + (−1.91 + 3.32i)9-s + (−2.75 − 1.58i)11-s + (2.72 + 4.71i)13-s + 0.259·15-s + 3.09i·17-s + (−1.45 − 0.838i)19-s − 4.40·21-s + (−5.97 − 3.45i)23-s + (2.49 + 4.32i)25-s − 2.18·27-s + (−1.50 − 0.871i)29-s + (−1.55 − 0.899i)31-s + ⋯ |
L(s) = 1 | + (0.754 + 1.30i)3-s + (0.0221 − 0.0383i)5-s + (−0.318 + 0.550i)7-s + (−0.639 + 1.10i)9-s + (−0.830 − 0.479i)11-s + (0.755 + 1.30i)13-s + 0.0669·15-s + 0.749i·17-s + (−0.333 − 0.192i)19-s − 0.960·21-s + (−1.24 − 0.719i)23-s + (0.499 + 0.864i)25-s − 0.421·27-s + (−0.280 − 0.161i)29-s + (−0.279 − 0.161i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 - 0.425i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.904 - 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.594976813\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.594976813\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 79 | \( 1 + (-6.15 + 6.40i)T \) |
good | 3 | \( 1 + (-1.30 - 2.26i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.0495 + 0.0858i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (0.841 - 1.45i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.75 + 1.58i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.72 - 4.71i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 3.09iT - 17T^{2} \) |
| 19 | \( 1 + (1.45 + 0.838i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (5.97 + 3.45i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.50 + 0.871i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1.55 + 0.899i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.60 - 2.08i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 3.90iT - 41T^{2} \) |
| 43 | \( 1 + (-1.98 - 3.44i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (0.0833 - 0.144i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.426 + 0.246i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4.28 + 7.42i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + 8.99iT - 61T^{2} \) |
| 67 | \( 1 - 15.2iT - 67T^{2} \) |
| 71 | \( 1 - 3.07T + 71T^{2} \) |
| 73 | \( 1 + (-0.753 + 1.30i)T + (-36.5 - 63.2i)T^{2} \) |
| 83 | \( 1 + (-5.11 - 2.95i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 1.59T + 89T^{2} \) |
| 97 | \( 1 - 15.6T + 97T^{2} \) |
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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
−9.886446330897646200398984351505, −9.205410294991622113934723295124, −8.610834800712229383566730708538, −7.984030311246792815069261533108, −6.58690147398430407752980028520, −5.78465800483910805853337040515, −4.72114063097936323618207677960, −3.93642681574157898033221065004, −3.12194824458847991704389033688, −2.02927292710755731118591105996,
0.58112370106047476429696827404, 1.94259166995694030658559108800, 2.88948343139979640057604130638, 3.82657251375931996309083535054, 5.26018093016248053905169236125, 6.18509658507351246874644534257, 7.11686923178371894501455801718, 7.69338950655955467432908175986, 8.260782029554398238898111836713, 9.143852871557377337404326180319