L(s) = 1 | + (1.80 − 1.31i)2-s + (−0.809 − 2.48i)3-s + (0.927 − 2.85i)4-s + (1.61 + 1.17i)5-s + (−4.73 − 3.44i)6-s + (−0.5 + 1.53i)7-s + (−0.690 − 2.12i)8-s + (−3.11 + 2.26i)9-s + 4.47·10-s + (−1.23 + 3.07i)11-s − 7.85·12-s + (−0.5 + 0.363i)13-s + (1.11 + 3.44i)14-s + (1.61 − 4.97i)15-s + (0.809 + 0.587i)16-s + (0.809 + 0.587i)17-s + ⋯ |
L(s) = 1 | + (1.27 − 0.929i)2-s + (−0.467 − 1.43i)3-s + (0.463 − 1.42i)4-s + (0.723 + 0.525i)5-s + (−1.93 − 1.40i)6-s + (−0.188 + 0.581i)7-s + (−0.244 − 0.751i)8-s + (−1.03 + 0.755i)9-s + 1.41·10-s + (−0.372 + 0.927i)11-s − 2.26·12-s + (−0.138 + 0.100i)13-s + (0.298 + 0.919i)14-s + (0.417 − 1.28i)15-s + (0.202 + 0.146i)16-s + (0.196 + 0.142i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.394 + 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.394 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.10736 - 1.68079i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.10736 - 1.68079i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (1.23 - 3.07i)T \) |
| 17 | \( 1 + (-0.809 - 0.587i)T \) |
good | 2 | \( 1 + (-1.80 + 1.31i)T + (0.618 - 1.90i)T^{2} \) |
| 3 | \( 1 + (0.809 + 2.48i)T + (-2.42 + 1.76i)T^{2} \) |
| 5 | \( 1 + (-1.61 - 1.17i)T + (1.54 + 4.75i)T^{2} \) |
| 7 | \( 1 + (0.5 - 1.53i)T + (-5.66 - 4.11i)T^{2} \) |
| 13 | \( 1 + (0.5 - 0.363i)T + (4.01 - 12.3i)T^{2} \) |
| 19 | \( 1 + (1.61 + 4.97i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + 3.85T + 23T^{2} \) |
| 29 | \( 1 + (-1.76 + 5.42i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (2 - 1.45i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (2 - 6.15i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-3.38 - 10.4i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 7.23T + 43T^{2} \) |
| 47 | \( 1 + (2.76 + 8.50i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (9.16 - 6.65i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-4.23 + 13.0i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (4.23 + 3.07i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 + (1.23 + 0.898i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-2.85 + 8.78i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (8.54 - 6.20i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-4.61 - 3.35i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 + (4.85 - 3.52i)T + (29.9 - 92.2i)T^{2} \) |
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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
−12.35788683367863206247164378438, −11.78477830283318200554688729198, −10.74918150174954255737677919299, −9.692457067752878927387821138186, −7.896405129844566490260414363222, −6.57248991350613759187906580815, −5.94300978934995928942665262725, −4.70436657882740417083427405519, −2.69673227750230445618789894744, −1.91891036566958514816456115725,
3.50842847506302592328034652341, 4.38884228502284554837322036563, 5.55564347229200479180294415659, 5.89254221262024890730657080009, 7.50793756582336497190414494125, 8.955164210184305362635593834380, 10.08345815291662001357877197296, 10.80991868388265130587499344605, 12.21678931208028670264386436181, 13.13592174582979061050916056079