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
\(L(1)\) |
\(\approx\) |
\(1.118921168\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.118921168\) |
\(L(\frac{3}{2})\) |
|
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
−9.710795084368439592660611234410, −8.861588001779673413166176064664, −7.69803875256133599697748984340, −7.53659377726064904703146185175, −6.18338791080535806340421838602, −5.54967921779778939136365101544, −4.86298696291929327071727648583, −3.20532421793221609955439359628, −2.29783317522426256336299654313, −0.77961725878771804556277850612,
0.859355846420349506124835871684, 3.18642936365321207950993252439, 3.50614338044103063868151081078, 4.78356325298883471557684678840, 5.47626420977099309981398029039, 6.72002477349140981434303610942, 7.13586454477864071972858856557, 8.318736188891231342334873067883, 9.334410413325694378516409114356, 10.00265994381525850672412155528