L(s) = 1 | + (−1.73 + 1.26i)2-s + (0.809 − 2.48i)4-s + (1.73 + 1.26i)5-s + (−1.30 + 4.02i)7-s + (0.410 + 1.26i)8-s − 4.61·10-s + (−3.22 + 0.780i)11-s + (−0.190 + 0.138i)13-s + (−2.81 − 8.65i)14-s + (1.92 + 1.40i)16-s + (4.96 + 3.60i)17-s + (−0.736 − 2.26i)19-s + (4.55 − 3.30i)20-s + (4.61 − 5.42i)22-s + 3.98·23-s + ⋯ |
L(s) = 1 | + (−1.22 + 0.893i)2-s + (0.404 − 1.24i)4-s + (0.777 + 0.564i)5-s + (−0.494 + 1.52i)7-s + (0.145 + 0.446i)8-s − 1.46·10-s + (−0.971 + 0.235i)11-s + (−0.0529 + 0.0384i)13-s + (−0.751 − 2.31i)14-s + (0.481 + 0.350i)16-s + (1.20 + 0.874i)17-s + (−0.168 − 0.519i)19-s + (1.01 − 0.739i)20-s + (0.984 − 1.15i)22-s + 0.830·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.503 - 0.863i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.503 - 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.283149 + 0.492927i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.283149 + 0.492927i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 + (3.22 - 0.780i)T \) |
good | 2 | \( 1 + (1.73 - 1.26i)T + (0.618 - 1.90i)T^{2} \) |
| 5 | \( 1 + (-1.73 - 1.26i)T + (1.54 + 4.75i)T^{2} \) |
| 7 | \( 1 + (1.30 - 4.02i)T + (-5.66 - 4.11i)T^{2} \) |
| 13 | \( 1 + (0.190 - 0.138i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-4.96 - 3.60i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (0.736 + 2.26i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 - 3.98T + 23T^{2} \) |
| 29 | \( 1 + (-2.14 + 6.61i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-3.73 + 2.71i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-0.545 + 1.67i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (0.917 + 2.82i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 2.70T + 43T^{2} \) |
| 47 | \( 1 + (-1.07 - 3.30i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-1.48 + 1.07i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (2.30 - 7.09i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-9.16 - 6.65i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + 2.85T + 67T^{2} \) |
| 71 | \( 1 + (8.69 + 6.31i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.763 + 2.35i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-7.66 + 5.56i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (6.70 + 4.86i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 - 8.90T + 89T^{2} \) |
| 97 | \( 1 + (9.54 - 6.93i)T + (29.9 - 92.2i)T^{2} \) |
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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
−14.79955660120720480422558176444, −13.25113797350179124783858718583, −12.16049313117143654646547169054, −10.47922753757290378885639170099, −9.748337247681020656371448084062, −8.788966779431701074935743152579, −7.72223354493215271637297475809, −6.33610167744056927485100486823, −5.62712327188243204286702406973, −2.57871003985398342127592857640,
1.07313258794831466183241338671, 3.13160208856049619505524521450, 5.24932177805359400080755508986, 7.16745378410721064430473741566, 8.277959486294670864035204387520, 9.582289154190474838956331460133, 10.17249319789931601031644645537, 10.97689273109095656707271204215, 12.40330400800730717211146210368, 13.33874906993058910753262045499