L(s) = 1 | + (1.69 + 0.348i)3-s + (−2.08 − 1.20i)5-s + (−2.53 + 0.763i)7-s + (2.75 + 1.18i)9-s + (2.81 − 1.62i)11-s + (4.35 − 2.51i)13-s + (−3.11 − 2.76i)15-s + (0.795 + 0.459i)17-s + (−3.22 − 5.57i)19-s + (−4.56 + 0.414i)21-s + (5.31 + 3.06i)23-s + (0.399 + 0.691i)25-s + (4.26 + 2.96i)27-s + (1.22 − 2.12i)29-s − 3.21·31-s + ⋯ |
L(s) = 1 | + (0.979 + 0.200i)3-s + (−0.932 − 0.538i)5-s + (−0.957 + 0.288i)7-s + (0.919 + 0.393i)9-s + (0.847 − 0.489i)11-s + (1.20 − 0.698i)13-s + (−0.805 − 0.714i)15-s + (0.192 + 0.111i)17-s + (−0.739 − 1.28i)19-s + (−0.995 + 0.0904i)21-s + (1.10 + 0.640i)23-s + (0.0798 + 0.138i)25-s + (0.821 + 0.570i)27-s + (0.228 − 0.394i)29-s − 0.578·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 + 0.631i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.775 + 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.862127082\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.862127082\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.69 - 0.348i)T \) |
| 7 | \( 1 + (2.53 - 0.763i)T \) |
good | 5 | \( 1 + (2.08 + 1.20i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.81 + 1.62i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-4.35 + 2.51i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.795 - 0.459i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.22 + 5.57i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.31 - 3.06i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.22 + 2.12i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 3.21T + 31T^{2} \) |
| 37 | \( 1 + (4.08 + 7.07i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-10.3 + 5.99i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-10.2 - 5.89i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 4.84T + 47T^{2} \) |
| 53 | \( 1 + (1.56 - 2.71i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 - 9.86T + 59T^{2} \) |
| 61 | \( 1 + 9.49iT - 61T^{2} \) |
| 67 | \( 1 + 0.359iT - 67T^{2} \) |
| 71 | \( 1 - 2.32iT - 71T^{2} \) |
| 73 | \( 1 + (4.01 + 2.31i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 4.36iT - 79T^{2} \) |
| 83 | \( 1 + (8.68 - 15.0i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (9.68 - 5.59i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (9.79 + 5.65i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.504818046881072046712833960428, −8.971352483538404743640181094155, −8.447114237599144418766808616522, −7.50897505780545298932861142638, −6.61943272066562973107138923241, −5.54618357456670770158452204189, −4.16624250031995026621033048207, −3.65581128099730966451725004796, −2.68039485443220685290081993218, −0.880686406788536001308225073715,
1.39105119408901264967193631228, 2.91081888119993540413547030055, 3.79233639457073278621093560449, 4.21703587078706922385359772307, 6.15651569738043430580854792063, 6.85899229222483060722730432562, 7.44653610340663223144207484397, 8.501391595559111350531047197556, 9.068265381942012404088719240803, 9.973603222845290078689768654758