L(s) = 1 | + 1.40i·3-s + (0.466 + 2.18i)5-s + 1.57i·7-s + 1.03·9-s + 4.35·11-s − 0.964i·13-s + (−3.06 + 0.655i)15-s − 0.300i·17-s + 8.62·19-s − 2.20·21-s + i·23-s + (−4.56 + 2.04i)25-s + 5.65i·27-s − 4.76·29-s − 5.59·31-s + ⋯ |
L(s) = 1 | + 0.810i·3-s + (0.208 + 0.977i)5-s + 0.593i·7-s + 0.343·9-s + 1.31·11-s − 0.267i·13-s + (−0.792 + 0.169i)15-s − 0.0728i·17-s + 1.97·19-s − 0.481·21-s + 0.208i·23-s + (−0.912 + 0.408i)25-s + 1.08i·27-s − 0.885·29-s − 1.00·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.208 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.208 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.16001 + 1.43388i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.16001 + 1.43388i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.466 - 2.18i)T \) |
| 23 | \( 1 - iT \) |
good | 3 | \( 1 - 1.40iT - 3T^{2} \) |
| 7 | \( 1 - 1.57iT - 7T^{2} \) |
| 11 | \( 1 - 4.35T + 11T^{2} \) |
| 13 | \( 1 + 0.964iT - 13T^{2} \) |
| 17 | \( 1 + 0.300iT - 17T^{2} \) |
| 19 | \( 1 - 8.62T + 19T^{2} \) |
| 29 | \( 1 + 4.76T + 29T^{2} \) |
| 31 | \( 1 + 5.59T + 31T^{2} \) |
| 37 | \( 1 + 4.38iT - 37T^{2} \) |
| 41 | \( 1 + 6.62T + 41T^{2} \) |
| 43 | \( 1 - 1.72iT - 43T^{2} \) |
| 47 | \( 1 + 0.687iT - 47T^{2} \) |
| 53 | \( 1 + 8.05iT - 53T^{2} \) |
| 59 | \( 1 + 5.74T + 59T^{2} \) |
| 61 | \( 1 + 13.6T + 61T^{2} \) |
| 67 | \( 1 - 6.49iT - 67T^{2} \) |
| 71 | \( 1 - 9.89T + 71T^{2} \) |
| 73 | \( 1 + 6.35iT - 73T^{2} \) |
| 79 | \( 1 - 6.95T + 79T^{2} \) |
| 83 | \( 1 - 0.185iT - 83T^{2} \) |
| 89 | \( 1 - 1.64T + 89T^{2} \) |
| 97 | \( 1 + 9.21iT - 97T^{2} \) |
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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
−10.19048539306740114358482836821, −9.476581145005718139902104880332, −9.073581247772755032648792438884, −7.61289822165355245071308653889, −6.97192250087386605156129975957, −5.91053216130758548966069325054, −5.12003973042275268428475889134, −3.80596566494850348908847509667, −3.22216692433979513559418067950, −1.71038154624222794229191579124,
1.01244124474159282961573048731, 1.72443413405064397042202014850, 3.54970418089357246667871999272, 4.47441359635681650101427716652, 5.52638102487388432751933904375, 6.55109411919356515492130044851, 7.31609792046464701545681251376, 8.002850965029145863743232473110, 9.228249838989303496357252843651, 9.505412819231422382856653575322