L(s) = 1 | + (−2.59 + 1.5i)3-s + (1.86 − 1.23i)5-s + (3 − 5.19i)9-s + (−1.5 − 2.59i)11-s − i·13-s + (−3 + 6i)15-s + (−4.33 + 2.5i)17-s + (−4 + 6.92i)19-s + (1.73 + i)23-s + (1.96 − 4.59i)25-s + 9i·27-s + 29-s + (1 + 1.73i)31-s + (7.79 + 4.5i)33-s + (−8.66 − 5i)37-s + ⋯ |
L(s) = 1 | + (−1.49 + 0.866i)3-s + (0.834 − 0.550i)5-s + (1 − 1.73i)9-s + (−0.452 − 0.783i)11-s − 0.277i·13-s + (−0.774 + 1.54i)15-s + (−1.05 + 0.606i)17-s + (−0.917 + 1.58i)19-s + (0.361 + 0.208i)23-s + (0.392 − 0.919i)25-s + 1.73i·27-s + 0.185·29-s + (0.179 + 0.311i)31-s + (1.35 + 0.783i)33-s + (−1.42 − 0.821i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.897 + 0.441i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.897 + 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0112457 - 0.0483779i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0112457 - 0.0483779i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.86 + 1.23i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (2.59 - 1.5i)T + (1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + iT - 13T^{2} \) |
| 17 | \( 1 + (4.33 - 2.5i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4 - 6.92i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.73 - i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - T + 29T^{2} \) |
| 31 | \( 1 + (-1 - 1.73i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (8.66 + 5i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + (9.52 + 5.5i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (5.19 - 3i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (5 + 8.66i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (8.66 - 5i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (-8.66 + 5i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (3.5 - 6.06i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + (-4 + 6.92i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 3iT - 97T^{2} \) |
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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
−9.964773640084573469199497970540, −8.949854535878068282689557785457, −8.197534895633158224916275998898, −6.58551052585754380153113085522, −6.05481972777082024739420643487, −5.32457013949904381309749558053, −4.61404768714028892667181800824, −3.53160333183484700010917107419, −1.68738957364225338642736906179, −0.02671729847966380313168690922,
1.69281579681639743066495187430, 2.63677986777861922652480865470, 4.70877252135584156490909615219, 5.14367182259279865371453991940, 6.41983240329714525238744977530, 6.69441759163130989770826021464, 7.37995952670290149355673365767, 8.733989035052104296055250331419, 9.739481295816638520365595440554, 10.63886509343875657704160256319