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
−9.950399104068475472946717090619, −9.227400435579643543248655821874, −7.87524522211378637907903114445, −7.73000005392820388020166851486, −6.40083396274494312508580814671, −5.82670881717135127554855943309, −4.94256569084432928456465695676, −3.77571055794531317520213630280, −2.85751783600073475192400976243, −1.68449042327469410319500197672,
0.05625481296129633888822167531, 2.11730044154648985538319091293, 2.71018020397887352198341340245, 3.94820100809520180916962892056, 5.33271233987586700360745523289, 5.62340988580215046908695956062, 6.59503479250635657680448570437, 7.59489109687059937584744173370, 8.292826708307665162854771526176, 9.325896338872743356762453578426