L(s) = 1 | + (−0.500 + 1.86i)2-s + (0.866 − 0.5i)3-s + (−1.50 − 0.870i)4-s + (−2.44 + 2.44i)5-s + (0.500 + 1.86i)6-s + (−2.60 − 0.461i)7-s + (−0.353 + 0.353i)8-s + (0.499 − 0.866i)9-s + (−3.33 − 5.78i)10-s + (2.85 + 0.764i)11-s − 1.74·12-s + (−3.60 − 0.0697i)13-s + (2.16 − 4.63i)14-s + (−0.893 + 3.33i)15-s + (−2.22 − 3.85i)16-s + (0.667 − 1.15i)17-s + ⋯ |
L(s) = 1 | + (−0.354 + 1.32i)2-s + (0.499 − 0.288i)3-s + (−0.754 − 0.435i)4-s + (−1.09 + 1.09i)5-s + (0.204 + 0.762i)6-s + (−0.984 − 0.174i)7-s + (−0.124 + 0.124i)8-s + (0.166 − 0.288i)9-s + (−1.05 − 1.82i)10-s + (0.860 + 0.230i)11-s − 0.502·12-s + (−0.999 − 0.0193i)13-s + (0.579 − 1.23i)14-s + (−0.230 + 0.860i)15-s + (−0.556 − 0.963i)16-s + (0.161 − 0.280i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 + 0.359i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.933 + 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.119528 - 0.643250i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.119528 - 0.643250i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 7 | \( 1 + (2.60 + 0.461i)T \) |
| 13 | \( 1 + (3.60 + 0.0697i)T \) |
good | 2 | \( 1 + (0.500 - 1.86i)T + (-1.73 - i)T^{2} \) |
| 5 | \( 1 + (2.44 - 2.44i)T - 5iT^{2} \) |
| 11 | \( 1 + (-2.85 - 0.764i)T + (9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (-0.667 + 1.15i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.32 - 4.94i)T + (-16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (7.61 - 4.39i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.97 - 6.89i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.04 + 2.04i)T - 31iT^{2} \) |
| 37 | \( 1 + (-4.73 - 1.26i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-1.45 - 0.390i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (0.212 + 0.122i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.13 - 1.13i)T + 47iT^{2} \) |
| 53 | \( 1 - 2.62T + 53T^{2} \) |
| 59 | \( 1 + (3.96 - 1.06i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (-7.82 - 4.52i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.97 - 11.1i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (11.0 - 2.96i)T + (61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (1.06 + 1.06i)T + 73iT^{2} \) |
| 79 | \( 1 - 7.65T + 79T^{2} \) |
| 83 | \( 1 + (-10.1 + 10.1i)T - 83iT^{2} \) |
| 89 | \( 1 + (4.11 - 15.3i)T + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (3.28 + 12.2i)T + (-84.0 + 48.5i)T^{2} \) |
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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
−12.22154987172280818332832354013, −11.76503495793216251709232845097, −10.17661988732162484588826664172, −9.415536472494400445186707093931, −8.151233588781192093236434340715, −7.40713568883620141306716473496, −6.88547142904639094835835466908, −5.93951991505800401007433775606, −4.02418900764227357656888779961, −2.91328114947600931927064767569,
0.51589159695663519836561875061, 2.51521523600154057233093992160, 3.75244096595694817063820632796, 4.53962447344688484073960954736, 6.42718112451686810352786967041, 7.915083523485997146966691466152, 8.878938630252348965525176695313, 9.504545559661002426392446752828, 10.31033075737206686016449784280, 11.63322145139792874387742023519