L(s) = 1 | + 1.58·5-s + (−1.80 + 1.93i)7-s − 5.17·11-s + (−0.681 + 1.18i)13-s + (2.30 − 3.99i)17-s + (0.0321 + 0.0557i)19-s − 6.74·23-s − 2.49·25-s + (−4.70 − 8.15i)29-s + (1.33 + 2.30i)31-s + (−2.86 + 3.05i)35-s + (0.880 + 1.52i)37-s + (0.858 − 1.48i)41-s + (−5.12 − 8.86i)43-s + (2.60 − 4.51i)47-s + ⋯ |
L(s) = 1 | + 0.707·5-s + (−0.683 + 0.729i)7-s − 1.55·11-s + (−0.189 + 0.327i)13-s + (0.559 − 0.969i)17-s + (0.00738 + 0.0127i)19-s − 1.40·23-s − 0.499·25-s + (−0.874 − 1.51i)29-s + (0.239 + 0.414i)31-s + (−0.483 + 0.516i)35-s + (0.144 + 0.250i)37-s + (0.134 − 0.232i)41-s + (−0.780 − 1.35i)43-s + (0.379 − 0.657i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.384i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.923 + 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1588580498\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1588580498\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1.80 - 1.93i)T \) |
good | 5 | \( 1 - 1.58T + 5T^{2} \) |
| 11 | \( 1 + 5.17T + 11T^{2} \) |
| 13 | \( 1 + (0.681 - 1.18i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.30 + 3.99i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.0321 - 0.0557i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 6.74T + 23T^{2} \) |
| 29 | \( 1 + (4.70 + 8.15i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.33 - 2.30i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.880 - 1.52i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.858 + 1.48i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (5.12 + 8.86i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.60 + 4.51i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.479 + 0.831i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4.66 + 8.08i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (7.19 - 12.4i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.24 - 10.8i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 4.49T + 71T^{2} \) |
| 73 | \( 1 + (0.941 - 1.63i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (3.26 - 5.65i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.08 - 8.81i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-4.12 - 7.14i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (7.26 + 12.5i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.325896338872743356762453578426, −8.292826708307665162854771526176, −7.59489109687059937584744173370, −6.59503479250635657680448570437, −5.62340988580215046908695956062, −5.33271233987586700360745523289, −3.94820100809520180916962892056, −2.71018020397887352198341340245, −2.11730044154648985538319091293, −0.05625481296129633888822167531,
1.68449042327469410319500197672, 2.85751783600073475192400976243, 3.77571055794531317520213630280, 4.94256569084432928456465695676, 5.82670881717135127554855943309, 6.40083396274494312508580814671, 7.73000005392820388020166851486, 7.87524522211378637907903114445, 9.227400435579643543248655821874, 9.950399104068475472946717090619