L(s) = 1 | + (1.40 − 0.171i)2-s + (1.94 − 0.481i)4-s − 0.963i·5-s + (2.64 − 1.00i)8-s + (−0.165 − 1.35i)10-s + 5.48i·11-s − 3.75i·13-s + (3.53 − 1.87i)16-s − 0.686i·17-s + 4.88·19-s + (−0.464 − 1.87i)20-s + (0.941 + 7.69i)22-s + 1.24i·23-s + 4.07·25-s + (−0.643 − 5.26i)26-s + ⋯ |
L(s) = 1 | + (0.992 − 0.121i)2-s + (0.970 − 0.240i)4-s − 0.430i·5-s + (0.934 − 0.356i)8-s + (−0.0523 − 0.427i)10-s + 1.65i·11-s − 1.04i·13-s + (0.883 − 0.467i)16-s − 0.166i·17-s + 1.12·19-s + (−0.103 − 0.418i)20-s + (0.200 + 1.64i)22-s + 0.258i·23-s + 0.814·25-s + (−0.126 − 1.03i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.891 + 0.453i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.891 + 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.590202555\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.590202555\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.40 + 0.171i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 0.963iT - 5T^{2} \) |
| 11 | \( 1 - 5.48iT - 11T^{2} \) |
| 13 | \( 1 + 3.75iT - 13T^{2} \) |
| 17 | \( 1 + 0.686iT - 17T^{2} \) |
| 19 | \( 1 - 4.88T + 19T^{2} \) |
| 23 | \( 1 - 1.24iT - 23T^{2} \) |
| 29 | \( 1 - 2.48T + 29T^{2} \) |
| 31 | \( 1 - 4.82T + 31T^{2} \) |
| 37 | \( 1 + 2.73T + 37T^{2} \) |
| 41 | \( 1 + 9.42iT - 41T^{2} \) |
| 43 | \( 1 + 5.97iT - 43T^{2} \) |
| 47 | \( 1 + 3.61T + 47T^{2} \) |
| 53 | \( 1 - 4.09T + 53T^{2} \) |
| 59 | \( 1 + 12.6T + 59T^{2} \) |
| 61 | \( 1 - 10.4iT - 61T^{2} \) |
| 67 | \( 1 + 9.43iT - 67T^{2} \) |
| 71 | \( 1 - 10.1iT - 71T^{2} \) |
| 73 | \( 1 - 6.66iT - 73T^{2} \) |
| 79 | \( 1 - 1.41iT - 79T^{2} \) |
| 83 | \( 1 + 0.543T + 83T^{2} \) |
| 89 | \( 1 + 0.554iT - 89T^{2} \) |
| 97 | \( 1 + 10.8iT - 97T^{2} \) |
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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
−9.428664090933407676234577570650, −8.303850883890047497165948496151, −7.36181900335552167243953940391, −6.93695186496066414671209656132, −5.72231409763471653469372591397, −5.07660514584728772978539118463, −4.41561752031387405947484246859, −3.33553128952847161929662951067, −2.38358945119011199434539036595, −1.17803378896638368248984709221,
1.32402907241858470187908466907, 2.82145891216457744108759035923, 3.32774851587226308112256179170, 4.43062738213065909221589903790, 5.24856336398397313818723153279, 6.30424521134577451710782025082, 6.56869843195796362303669849184, 7.68408301038359337759567910459, 8.380468002996167887430444267477, 9.336126832414285716799290633978