L(s) = 1 | + (−0.989 + 0.142i)2-s + (−0.540 + 0.841i)3-s + (0.959 − 0.281i)4-s + (0.909 + 0.415i)5-s + (0.415 − 0.909i)6-s + (−0.909 + 0.415i)8-s + (−0.415 − 0.909i)9-s + (−0.959 − 0.281i)10-s + (−0.281 + 0.959i)12-s + (−0.841 + 0.540i)15-s + (0.841 − 0.540i)16-s + (−0.281 + 0.0405i)17-s + (0.540 + 0.841i)18-s + (−0.273 + 1.89i)19-s + (0.989 + 0.142i)20-s + ⋯ |
L(s) = 1 | + (−0.989 + 0.142i)2-s + (−0.540 + 0.841i)3-s + (0.959 − 0.281i)4-s + (0.909 + 0.415i)5-s + (0.415 − 0.909i)6-s + (−0.909 + 0.415i)8-s + (−0.415 − 0.909i)9-s + (−0.959 − 0.281i)10-s + (−0.281 + 0.959i)12-s + (−0.841 + 0.540i)15-s + (0.841 − 0.540i)16-s + (−0.281 + 0.0405i)17-s + (0.540 + 0.841i)18-s + (−0.273 + 1.89i)19-s + (0.989 + 0.142i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.167 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.167 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6490885706\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6490885706\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.989 - 0.142i)T \) |
| 3 | \( 1 + (0.540 - 0.841i)T \) |
| 5 | \( 1 + (-0.909 - 0.415i)T \) |
| 23 | \( 1 + (-0.755 - 0.654i)T \) |
good | 7 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 11 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 13 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 17 | \( 1 + (0.281 - 0.0405i)T + (0.959 - 0.281i)T^{2} \) |
| 19 | \( 1 + (0.273 - 1.89i)T + (-0.959 - 0.281i)T^{2} \) |
| 29 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 31 | \( 1 + (0.304 + 0.474i)T + (-0.415 + 0.909i)T^{2} \) |
| 37 | \( 1 + (-0.654 + 0.755i)T^{2} \) |
| 41 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 43 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 47 | \( 1 + 1.30iT - T^{2} \) |
| 53 | \( 1 + (0.368 - 1.25i)T + (-0.841 - 0.540i)T^{2} \) |
| 59 | \( 1 + (0.841 - 0.540i)T^{2} \) |
| 61 | \( 1 + (-0.817 - 1.27i)T + (-0.415 + 0.909i)T^{2} \) |
| 67 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 71 | \( 1 + (-0.142 + 0.989i)T^{2} \) |
| 73 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 79 | \( 1 + (0.797 - 0.234i)T + (0.841 - 0.540i)T^{2} \) |
| 83 | \( 1 + (0.449 + 0.983i)T + (-0.654 + 0.755i)T^{2} \) |
| 89 | \( 1 + (0.415 + 0.909i)T^{2} \) |
| 97 | \( 1 + (-0.654 - 0.755i)T^{2} \) |
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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
−10.11922651951114379486470018554, −9.293488930656173356014103783680, −8.703960620952681225346660109023, −7.58797164083192641153282888584, −6.67743264505803140412515251215, −5.87352072365062387268150291821, −5.40551909832488514069668983977, −3.93057625382645842199638544767, −2.81667026418394936151930437020, −1.52685587314587418990601557877,
0.832052279901444240169861029817, 2.04033792919189885890668540913, 2.83030027337197572264875632950, 4.74206593006363608855338267104, 5.60431657858491071430970387892, 6.68154489916735442246129993477, 6.88448903244348851645852945396, 8.067468863729801989359653043817, 8.818762466343192270300341109479, 9.391887809397564942643296414243