L(s) = 1 | + (−0.809 − 0.587i)2-s + (0.118 − 0.363i)3-s + (0.309 + 0.951i)4-s + (−2.61 + 1.90i)5-s + (−0.309 + 0.224i)6-s + (−0.618 − 1.90i)7-s + (0.309 − 0.951i)8-s + (2.30 + 1.67i)9-s + 3.23·10-s + (0.309 − 3.30i)11-s + 0.381·12-s + (−1 − 0.726i)13-s + (−0.618 + 1.90i)14-s + (0.381 + 1.17i)15-s + (−0.809 + 0.587i)16-s + (0.5 − 0.363i)17-s + ⋯ |
L(s) = 1 | + (−0.572 − 0.415i)2-s + (0.0681 − 0.209i)3-s + (0.154 + 0.475i)4-s + (−1.17 + 0.850i)5-s + (−0.126 + 0.0916i)6-s + (−0.233 − 0.718i)7-s + (0.109 − 0.336i)8-s + (0.769 + 0.559i)9-s + 1.02·10-s + (0.0931 − 0.995i)11-s + 0.110·12-s + (−0.277 − 0.201i)13-s + (−0.165 + 0.508i)14-s + (0.0986 + 0.303i)15-s + (−0.202 + 0.146i)16-s + (0.121 − 0.0881i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.459919 - 0.0893326i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.459919 - 0.0893326i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.809 + 0.587i)T \) |
| 11 | \( 1 + (-0.309 + 3.30i)T \) |
good | 3 | \( 1 + (-0.118 + 0.363i)T + (-2.42 - 1.76i)T^{2} \) |
| 5 | \( 1 + (2.61 - 1.90i)T + (1.54 - 4.75i)T^{2} \) |
| 7 | \( 1 + (0.618 + 1.90i)T + (-5.66 + 4.11i)T^{2} \) |
| 13 | \( 1 + (1 + 0.726i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.5 + 0.363i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (1.80 - 5.56i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 - 1.23T + 23T^{2} \) |
| 29 | \( 1 + (-1.38 - 4.25i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (1.61 + 1.17i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (1.14 + 3.52i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-1.73 + 5.34i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 8.56T + 43T^{2} \) |
| 47 | \( 1 + (2 - 6.15i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-1.23 - 0.898i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (2.66 + 8.19i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-2 + 1.45i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 - 11.0T + 67T^{2} \) |
| 71 | \( 1 + (-4.23 + 3.07i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-3.20 - 9.87i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (10.8 + 7.88i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (7.54 - 5.48i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + 8.09T + 89T^{2} \) |
| 97 | \( 1 + (5.78 + 4.20i)T + (29.9 + 92.2i)T^{2} \) |
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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
−18.53323601488478753406428324084, −16.77678219891911949871257985517, −15.78808119532245846683109328974, −14.21927936304491680625721920771, −12.65389133375843142327103425648, −11.19015273008704886636155042019, −10.26581297587051140598940030101, −8.119000129251193742156669325387, −7.05504734034573282193115387091, −3.65206342848791883968556909270,
4.60558888920413881633024178031, 7.02176996947940726846070608250, 8.578299937453888876233453362782, 9.732900602864643249518309140274, 11.73196195975361073076144697363, 12.78371803078076126247183040055, 15.16179757497127178864373049691, 15.54777422558285971247884950111, 16.82411981911449626733863598771, 18.17756986172014387027382435622