L(s) = 1 | + (0.857 − 1.50i)3-s + (−0.551 + 0.955i)5-s + (1.62 + 2.81i)7-s + (−1.53 − 2.58i)9-s + (1.28 + 2.23i)11-s + (−1.58 + 2.74i)13-s + (0.965 + 1.64i)15-s + 4.71·17-s − 5.75·19-s + (5.62 − 0.0333i)21-s + (−2.35 + 4.07i)23-s + (1.89 + 3.27i)25-s + (−5.19 + 0.0922i)27-s + (3.66 + 6.34i)29-s + (2.93 − 5.07i)31-s + ⋯ |
L(s) = 1 | + (0.494 − 0.868i)3-s + (−0.246 + 0.427i)5-s + (0.614 + 1.06i)7-s + (−0.510 − 0.860i)9-s + (0.388 + 0.672i)11-s + (−0.440 + 0.762i)13-s + (0.249 + 0.425i)15-s + 1.14·17-s − 1.32·19-s + (1.22 − 0.00726i)21-s + (−0.490 + 0.850i)23-s + (0.378 + 0.655i)25-s + (−0.999 + 0.0177i)27-s + (0.680 + 1.17i)29-s + (0.526 − 0.911i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.758 - 0.651i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.758 - 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.803154835\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.803154835\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.857 + 1.50i)T \) |
good | 5 | \( 1 + (0.551 - 0.955i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-1.62 - 2.81i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.28 - 2.23i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.58 - 2.74i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 4.71T + 17T^{2} \) |
| 19 | \( 1 + 5.75T + 19T^{2} \) |
| 23 | \( 1 + (2.35 - 4.07i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.66 - 6.34i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.93 + 5.07i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 0.0714T + 37T^{2} \) |
| 41 | \( 1 + (1.63 - 2.83i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.12 + 3.67i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.72 - 8.18i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 6.42T + 53T^{2} \) |
| 59 | \( 1 + (4.19 - 7.26i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.66 + 8.07i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.09 + 10.5i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 0.335T + 71T^{2} \) |
| 73 | \( 1 - 14.8T + 73T^{2} \) |
| 79 | \( 1 + (4.85 + 8.41i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3.07 + 5.31i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 4.42T + 89T^{2} \) |
| 97 | \( 1 + (-6.39 - 11.0i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.633103807869016450553740588139, −8.997807841714011219401070481383, −8.174899144023065662200062464202, −7.46795004227256258870599657091, −6.67861663448098396656795340580, −5.83865443304354566164212723188, −4.73264058368747074914743491477, −3.51155214945417704796643957605, −2.40362731998595226320756429627, −1.58443556696705514682940285414,
0.77128459380191517537383994445, 2.50961216434754595867452236041, 3.71228234404731670640784449743, 4.37793674661880900960581853606, 5.16100300845831209827500956229, 6.28727099455897240866324220996, 7.49808691979617511573029179823, 8.411359306922882996787798992753, 8.515864874117456359961978134429, 10.02560201651576733258294823113