L(s) = 1 | + (−1.38 + 0.285i)2-s − 3-s + (1.83 − 0.791i)4-s + 2.03·5-s + (1.38 − 0.285i)6-s − 1.21·7-s + (−2.31 + 1.62i)8-s + 9-s + (−2.81 + 0.580i)10-s + 1.88i·11-s + (−1.83 + 0.791i)12-s + 3.19i·13-s + (1.68 − 0.348i)14-s − 2.03·15-s + (2.74 − 2.90i)16-s + 5.91i·17-s + ⋯ |
L(s) = 1 | + (−0.979 + 0.201i)2-s − 0.577·3-s + (0.918 − 0.395i)4-s + 0.909·5-s + (0.565 − 0.116i)6-s − 0.461·7-s + (−0.819 + 0.572i)8-s + 0.333·9-s + (−0.890 + 0.183i)10-s + 0.569i·11-s + (−0.530 + 0.228i)12-s + 0.884i·13-s + (0.451 − 0.0931i)14-s − 0.524·15-s + (0.687 − 0.726i)16-s + 1.43i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.355 - 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.355 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.655134 + 0.451587i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.655134 + 0.451587i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 + (-3.96 + 2.69i)T \) |
good | 5 | \( 1 - 2.03T + 5T^{2} \) |
| 7 | \( 1 + 1.21T + 7T^{2} \) |
| 11 | \( 1 - 1.88iT - 11T^{2} \) |
| 13 | \( 1 - 3.19iT - 13T^{2} \) |
| 17 | \( 1 - 5.91iT - 17T^{2} \) |
| 19 | \( 1 + 6.08iT - 19T^{2} \) |
| 29 | \( 1 - 9.30iT - 29T^{2} \) |
| 31 | \( 1 + 3.55iT - 31T^{2} \) |
| 37 | \( 1 - 11.1T + 37T^{2} \) |
| 41 | \( 1 + 2.52T + 41T^{2} \) |
| 43 | \( 1 - 11.6iT - 43T^{2} \) |
| 47 | \( 1 - 11.2iT - 47T^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 59 | \( 1 - 0.657T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 - 2.45iT - 67T^{2} \) |
| 71 | \( 1 - 4.84iT - 71T^{2} \) |
| 73 | \( 1 + 7.87T + 73T^{2} \) |
| 79 | \( 1 + 4.23T + 79T^{2} \) |
| 83 | \( 1 - 7.58iT - 83T^{2} \) |
| 89 | \( 1 + 3.85iT - 89T^{2} \) |
| 97 | \( 1 + 3.03iT - 97T^{2} \) |
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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
−10.87458532572843233837404864094, −9.868366239643979392835040375255, −9.406305161878149597627276001315, −8.484151673371134673720693488267, −7.16607153638307145663730122748, −6.51984124367890210652575512180, −5.78849693181712488167249873311, −4.53911096853769708330135901756, −2.64686867289233235109520438015, −1.39485347919611959760566325306,
0.72385252080137724555413810285, 2.31650009025661417909940376068, 3.53919384112991330808014511654, 5.43843250849313849678714384782, 6.04230946489857678662748144237, 7.06221523645040071790805774180, 7.996360923784416397992620423245, 9.070302280120135263013018863669, 9.972764993901618755951365904238, 10.24522528085115891039039771687