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
−10.02560201651576733258294823113, −8.515864874117456359961978134429, −8.411359306922882996787798992753, −7.49808691979617511573029179823, −6.28727099455897240866324220996, −5.16100300845831209827500956229, −4.37793674661880900960581853606, −3.71228234404731670640784449743, −2.50961216434754595867452236041, −0.77128459380191517537383994445,
1.58443556696705514682940285414, 2.40362731998595226320756429627, 3.51155214945417704796643957605, 4.73264058368747074914743491477, 5.83865443304354566164212723188, 6.67861663448098396656795340580, 7.46795004227256258870599657091, 8.174899144023065662200062464202, 8.997807841714011219401070481383, 9.633103807869016450553740588139