L(s) = 1 | + (−1.38 + 0.285i)2-s − i·3-s + (1.83 − 0.791i)4-s + 2.43i·5-s + (0.285 + 1.38i)6-s − 3.50·7-s + (−2.31 + 1.62i)8-s − 9-s + (−0.696 − 3.37i)10-s − 2.10i·11-s + (−0.791 − 1.83i)12-s − 1.86i·13-s + (4.85 − 1.00i)14-s + 2.43·15-s + (2.74 − 2.90i)16-s + 7.45·17-s + ⋯ |
L(s) = 1 | + (−0.979 + 0.202i)2-s − 0.577i·3-s + (0.918 − 0.395i)4-s + 1.09i·5-s + (0.116 + 0.565i)6-s − 1.32·7-s + (−0.819 + 0.573i)8-s − 0.333·9-s + (−0.220 − 1.06i)10-s − 0.634i·11-s + (−0.228 − 0.530i)12-s − 0.517i·13-s + (1.29 − 0.267i)14-s + 0.629·15-s + (0.686 − 0.726i)16-s + 1.80·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.573 + 0.819i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.573 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.643274 - 0.335133i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.643274 - 0.335133i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.38 - 0.285i)T \) |
| 3 | \( 1 + iT \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 - 2.43iT - 5T^{2} \) |
| 7 | \( 1 + 3.50T + 7T^{2} \) |
| 11 | \( 1 + 2.10iT - 11T^{2} \) |
| 13 | \( 1 + 1.86iT - 13T^{2} \) |
| 17 | \( 1 - 7.45T + 17T^{2} \) |
| 19 | \( 1 + 6.83iT - 19T^{2} \) |
| 29 | \( 1 + 1.91iT - 29T^{2} \) |
| 31 | \( 1 - 3.05T + 31T^{2} \) |
| 37 | \( 1 + 6.64iT - 37T^{2} \) |
| 41 | \( 1 - 8.85T + 41T^{2} \) |
| 43 | \( 1 + 1.33iT - 43T^{2} \) |
| 47 | \( 1 + 0.423T + 47T^{2} \) |
| 53 | \( 1 - 5.74iT - 53T^{2} \) |
| 59 | \( 1 + 6.88iT - 59T^{2} \) |
| 61 | \( 1 + 5.52iT - 61T^{2} \) |
| 67 | \( 1 - 5.44iT - 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 + 3.86T + 73T^{2} \) |
| 79 | \( 1 - 7.55T + 79T^{2} \) |
| 83 | \( 1 + 16.3iT - 83T^{2} \) |
| 89 | \( 1 - 0.143T + 89T^{2} \) |
| 97 | \( 1 + 18.0T + 97T^{2} \) |
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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
−10.55501063654419262092415865845, −9.765311412781178423733924986783, −8.959313688058972081339263359510, −7.78169125916017783623285967296, −7.13825590728282576015255900173, −6.34299782467248598049350087018, −5.65818092177363031592585127880, −3.22528035119319522019077207044, −2.71452004356910948919169774135, −0.64829939520379934221487467465,
1.25600729382888655640740342592, 3.01921988311558869095208047004, 4.04239160296003436869744081199, 5.48038324540526768711186518223, 6.42431243464516097930174233253, 7.61299456977817506596833996232, 8.465604171410707015106907375567, 9.437260815635088778686922182150, 9.812788956976299105443044914643, 10.49295733884374519862715221231