L(s) = 1 | + (0.0709 + 1.73i)3-s + (2.04 + 3.54i)5-s + (−2.21 − 1.44i)7-s + (−2.98 + 0.245i)9-s + (3.24 + 1.87i)11-s + (2.68 + 1.54i)13-s + (−5.99 + 3.79i)15-s + (−0.219 − 0.379i)17-s + (−2.68 − 1.54i)19-s + (2.34 − 3.93i)21-s + (−2.43 + 1.40i)23-s + (−5.90 + 10.2i)25-s + (−0.636 − 5.15i)27-s + (0.122 − 0.0704i)29-s − 10.1i·31-s + ⋯ |
L(s) = 1 | + (0.0409 + 0.999i)3-s + (0.916 + 1.58i)5-s + (−0.836 − 0.547i)7-s + (−0.996 + 0.0818i)9-s + (0.979 + 0.565i)11-s + (0.743 + 0.429i)13-s + (−1.54 + 0.980i)15-s + (−0.0531 − 0.0920i)17-s + (−0.615 − 0.355i)19-s + (0.512 − 0.858i)21-s + (−0.508 + 0.293i)23-s + (−1.18 + 2.04i)25-s + (−0.122 − 0.992i)27-s + (0.0226 − 0.0130i)29-s − 1.81i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.562 - 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.562 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.693177 + 1.31068i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.693177 + 1.31068i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.0709 - 1.73i)T \) |
| 7 | \( 1 + (2.21 + 1.44i)T \) |
good | 5 | \( 1 + (-2.04 - 3.54i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-3.24 - 1.87i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.68 - 1.54i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (0.219 + 0.379i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.68 + 1.54i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.43 - 1.40i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.122 + 0.0704i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 10.1iT - 31T^{2} \) |
| 37 | \( 1 + (1.72 - 2.98i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.78 + 3.09i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.66 - 8.08i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 5.53T + 47T^{2} \) |
| 53 | \( 1 + (-7.08 + 4.09i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 - 9.02T + 59T^{2} \) |
| 61 | \( 1 - 15.0iT - 61T^{2} \) |
| 67 | \( 1 + 6.30T + 67T^{2} \) |
| 71 | \( 1 + 0.881iT - 71T^{2} \) |
| 73 | \( 1 + (-10.3 + 5.95i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 - 3.29T + 79T^{2} \) |
| 83 | \( 1 + (0.293 + 0.508i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-7.77 + 13.4i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-8.29 + 4.78i)T + (48.5 - 84.0i)T^{2} \) |
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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
−11.04037391012214911447382865742, −10.16475888031622106160533856635, −9.738348576464805438884074782796, −8.950085558911454171425673399576, −7.37738084572358219404194678715, −6.41416176463821647666027645824, −5.97391350241165721403872917307, −4.24282791837695875111337198995, −3.46622998355586688442020356386, −2.29905803712603227387102810650,
0.914303258132455530866634033774, 2.07653288710482082641543847311, 3.65030070081075299941573713870, 5.26850152313465965319254239243, 6.02676318300877901491238263938, 6.63337534787142425360050916815, 8.280355552625605118170094666288, 8.749478324095255607506073625730, 9.383315763137330455743673932759, 10.57536121855573644837796678685