L(s) = 1 | + (1.20 + 2.09i)3-s + (−1.06 − 1.84i)5-s − 1.59·7-s + (−1.41 + 2.44i)9-s + 1.12·11-s + (3.27 − 5.67i)13-s + (2.56 − 4.45i)15-s + (−2.14 − 3.72i)17-s + (2.69 − 3.42i)19-s + (−1.92 − 3.33i)21-s + (−2.85 + 4.94i)23-s + (0.233 − 0.404i)25-s + 0.414·27-s + (0.882 − 1.52i)29-s + 5.25·31-s + ⋯ |
L(s) = 1 | + (0.696 + 1.20i)3-s + (−0.476 − 0.824i)5-s − 0.603·7-s + (−0.471 + 0.816i)9-s + 0.340·11-s + (0.908 − 1.57i)13-s + (0.663 − 1.14i)15-s + (−0.520 − 0.902i)17-s + (0.617 − 0.786i)19-s + (−0.420 − 0.728i)21-s + (−0.595 + 1.03i)23-s + (0.0467 − 0.0809i)25-s + 0.0797·27-s + (0.163 − 0.283i)29-s + 0.943·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 + 0.347i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.937 + 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.755827861\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.755827861\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (-2.69 + 3.42i)T \) |
good | 3 | \( 1 + (-1.20 - 2.09i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (1.06 + 1.84i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + 1.59T + 7T^{2} \) |
| 11 | \( 1 - 1.12T + 11T^{2} \) |
| 13 | \( 1 + (-3.27 + 5.67i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (2.14 + 3.72i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (2.85 - 4.94i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.882 + 1.52i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 5.25T + 31T^{2} \) |
| 37 | \( 1 - 1.23T + 37T^{2} \) |
| 41 | \( 1 + (-4.04 - 7.00i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.35 + 4.08i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.973 + 1.68i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.56 + 7.90i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.33 - 2.31i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5.89 - 10.2i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.87 + 10.1i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.71 - 4.70i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.967 - 1.67i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (7.49 + 12.9i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 4.06T + 83T^{2} \) |
| 89 | \( 1 + (-0.680 + 1.17i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (9.00 + 15.5i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.708955931585857847260835652739, −8.861123868368239735503968584460, −8.379583606983479787986932989580, −7.46408288927450835902735522093, −6.21254439782630563136739484338, −5.17433573668628605987393712898, −4.42629633650535298281055107512, −3.51257378547856897771558067468, −2.85084389742012291569812459291, −0.75663905149875832281039025593,
1.40034248160446331180875069100, 2.43315907263584100692859734101, 3.48413169980041928593711717316, 4.26715230642203893102455239693, 6.14189035115509850990292189610, 6.57872659928469507595561036935, 7.20031770721882813243207844850, 8.143331444735239822970156412410, 8.721955682077209871415254579185, 9.641159621307258439804831715452