L(s) = 1 | + (−0.427 − 0.247i)2-s + (−0.866 + 0.5i)3-s + (−0.877 − 1.52i)4-s + 0.494·6-s + (1.86 − 1.87i)7-s + 1.85i·8-s + (0.499 − 0.866i)9-s + (−2.66 − 4.62i)11-s + (1.52 + 0.877i)12-s + 5.09i·13-s + (−1.26 + 0.343i)14-s + (−1.29 + 2.24i)16-s + (−0.303 + 0.175i)17-s + (−0.427 + 0.247i)18-s + (1.38 − 2.39i)19-s + ⋯ |
L(s) = 1 | + (−0.302 − 0.174i)2-s + (−0.499 + 0.288i)3-s + (−0.438 − 0.760i)4-s + 0.201·6-s + (0.704 − 0.709i)7-s + 0.656i·8-s + (0.166 − 0.288i)9-s + (−0.804 − 1.39i)11-s + (0.438 + 0.253i)12-s + 1.41i·13-s + (−0.337 + 0.0917i)14-s + (−0.324 + 0.561i)16-s + (−0.0735 + 0.0424i)17-s + (−0.100 + 0.0582i)18-s + (0.317 − 0.549i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.925 + 0.379i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.925 + 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0937085 - 0.475813i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0937085 - 0.475813i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.866 - 0.5i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.86 + 1.87i)T \) |
good | 2 | \( 1 + (0.427 + 0.247i)T + (1 + 1.73i)T^{2} \) |
| 11 | \( 1 + (2.66 + 4.62i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 5.09iT - 13T^{2} \) |
| 17 | \( 1 + (0.303 - 0.175i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.38 + 2.39i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (6.50 + 3.75i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 + (3.05 + 5.29i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.05 + 1.76i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 7.86T + 41T^{2} \) |
| 43 | \( 1 - 1.41iT - 43T^{2} \) |
| 47 | \( 1 + (7.66 + 4.42i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (5.47 - 3.16i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.42 - 11.1i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.5 + 4.33i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.25 - 1.87i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 15.4T + 71T^{2} \) |
| 73 | \( 1 + (-6.64 + 3.83i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.18 + 2.04i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 6.87iT - 83T^{2} \) |
| 89 | \( 1 + (2.17 - 3.76i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 5.49iT - 97T^{2} \) |
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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
−10.55890887822231742897408910303, −9.735783366705447880422345015424, −8.793333838410920831136935612404, −7.982389321692956927354449835618, −6.67348041954034412107091766964, −5.69035377379757419486319198383, −4.83124262164406627866059058222, −3.87193879019630366847326106328, −1.90744400506504890934318377216, −0.32998435000809165745965217395,
1.95505052119967522859861589267, 3.45643359408414031323207791208, 4.90444329045725408182430888313, 5.49231036875446968177084439346, 6.93760159739870542369784633824, 7.934814400855307067569185780826, 8.151884732007460517624924369741, 9.562660791344155221568773545719, 10.20831478516013547416818075171, 11.34716955647624215338641318625