L(s) = 1 | + (−1.20 − 2.09i)3-s + (0.292 − 0.507i)5-s + (−1.62 − 2.09i)7-s + (−1.41 + 2.44i)9-s + (0.5 + 0.866i)11-s − 3.82·13-s − 1.41·15-s + (−1.82 − 3.16i)17-s + (−0.292 + 0.507i)19-s + (−2.41 + 5.91i)21-s + (−3.12 + 5.40i)23-s + (2.32 + 4.03i)25-s − 0.414·27-s + 2.65·29-s + (−2 − 3.46i)31-s + ⋯ |
L(s) = 1 | + (−0.696 − 1.20i)3-s + (0.130 − 0.226i)5-s + (−0.612 − 0.790i)7-s + (−0.471 + 0.816i)9-s + (0.150 + 0.261i)11-s − 1.06·13-s − 0.365·15-s + (−0.443 − 0.768i)17-s + (−0.0671 + 0.116i)19-s + (−0.526 + 1.29i)21-s + (−0.650 + 1.12i)23-s + (0.465 + 0.806i)25-s − 0.0797·27-s + 0.493·29-s + (−0.359 − 0.622i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1002383163\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1002383163\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (1.62 + 2.09i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
good | 3 | \( 1 + (1.20 + 2.09i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.292 + 0.507i)T + (-2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 + 3.82T + 13T^{2} \) |
| 17 | \( 1 + (1.82 + 3.16i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.292 - 0.507i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.12 - 5.40i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 2.65T + 29T^{2} \) |
| 31 | \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.70 + 8.15i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 5.41T + 41T^{2} \) |
| 43 | \( 1 - 5.65T + 43T^{2} \) |
| 47 | \( 1 + (5.24 - 9.08i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3.94 + 6.84i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.79 + 4.83i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5.91 - 10.2i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.37 - 2.38i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 + (-4.70 - 8.15i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (6.62 - 11.4i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 + (6.24 - 10.8i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 3.82T + 97T^{2} \) |
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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
−9.433911468092206656512134201754, −7.941170254620526812192221513244, −7.30365044964073052401500454316, −6.80526593034520889293757107562, −5.90937410513882109939019523014, −5.02463484823249842840674078748, −3.89926753677861942613400968106, −2.50964540095253481216939716324, −1.28972389525511750117094385503, −0.04769916397000864123673790202,
2.30616989307676701334637295369, 3.35806701696570337299925675011, 4.50281456137945828038769451159, 5.08140903004630123525536019630, 6.18855204999598024099276178240, 6.57415811418307677688208446974, 8.037247626090067700191743454572, 8.905676559841936710913831968250, 9.647374266896851547638980251922, 10.33976535336361511477772731819