L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (−1.5 − 2.59i)5-s + (2.5 + 0.866i)7-s + 0.999·8-s + (−1.5 + 2.59i)10-s + (−0.5 + 0.866i)11-s + 2·13-s + (−0.500 − 2.59i)14-s + (−0.5 − 0.866i)16-s + (1.5 − 2.59i)17-s + (−1 − 1.73i)19-s + 3·20-s + 0.999·22-s + (1.5 + 2.59i)23-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.670 − 1.16i)5-s + (0.944 + 0.327i)7-s + 0.353·8-s + (−0.474 + 0.821i)10-s + (−0.150 + 0.261i)11-s + 0.554·13-s + (−0.133 − 0.694i)14-s + (−0.125 − 0.216i)16-s + (0.363 − 0.630i)17-s + (−0.229 − 0.397i)19-s + 0.670·20-s + 0.213·22-s + (0.312 + 0.541i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.257285605\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.257285605\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.5 - 0.866i)T \) |
| 11 | \( 1 + (0.5 - 0.866i)T \) |
good | 5 | \( 1 + (1.5 + 2.59i)T + (-2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 + (4.5 + 7.79i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (6 - 10.3i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.5 + 4.33i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.5 - 4.33i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + (-8 + 13.8i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (8.5 + 14.7i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 9T + 83T^{2} \) |
| 89 | \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 17T + 97T^{2} \) |
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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
−9.106582593319319140327175679174, −8.646188225349877601386588116476, −7.960369256014594624864973877121, −7.24094646244902765700639984972, −5.78715347848264763768852222171, −4.79876517842271712229745900127, −4.34670305938235612015213356893, −3.07480895586425640386497719543, −1.76966996698982048694031002079, −0.68392592750658741446396216996,
1.25211510885093977073377401536, 2.81638326254037990007750585255, 3.89553828636340115483740286384, 4.78046277483799343820175957142, 5.94864695055717207207822762676, 6.67322426898016020530935443680, 7.47037256122227930271297546017, 8.150354029420569066497710908951, 8.659725080321463544619447621246, 9.949354119307875869852054270915