L(s) = 1 | − i·2-s + (−1.42 − 0.988i)3-s − 4-s + (1.41 + 2.44i)5-s + (−0.988 + 1.42i)6-s + (0.383 − 2.61i)7-s + i·8-s + (1.04 + 2.81i)9-s + (2.44 − 1.41i)10-s + (−4.64 + 2.68i)11-s + (1.42 + 0.988i)12-s + (−2.04 + 2.97i)13-s + (−2.61 − 0.383i)14-s + (0.408 − 4.88i)15-s + 16-s − 7.32·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (−0.821 − 0.570i)3-s − 0.5·4-s + (0.632 + 1.09i)5-s + (−0.403 + 0.580i)6-s + (0.145 − 0.989i)7-s + 0.353i·8-s + (0.349 + 0.937i)9-s + (0.774 − 0.447i)10-s + (−1.40 + 0.808i)11-s + (0.410 + 0.285i)12-s + (−0.565 + 0.824i)13-s + (−0.699 − 0.102i)14-s + (0.105 − 1.26i)15-s + 0.250·16-s − 1.77·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.433 - 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.433 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.495241 + 0.311291i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.495241 + 0.311291i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + (1.42 + 0.988i)T \) |
| 7 | \( 1 + (-0.383 + 2.61i)T \) |
| 13 | \( 1 + (2.04 - 2.97i)T \) |
good | 5 | \( 1 + (-1.41 - 2.44i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (4.64 - 2.68i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + 7.32T + 17T^{2} \) |
| 19 | \( 1 + (-2.17 - 1.25i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 4.84iT - 23T^{2} \) |
| 29 | \( 1 + (-5.21 - 3.01i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.76 - 1.01i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 0.639T + 37T^{2} \) |
| 41 | \( 1 + (5.97 - 10.3i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.836 + 1.44i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (2.48 + 4.31i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.82 - 1.63i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 4.67T + 59T^{2} \) |
| 61 | \( 1 + (-4.84 - 2.79i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.19 + 3.79i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-7.14 + 4.12i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.55 - 0.896i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.02 - 3.51i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 + 16.2T + 89T^{2} \) |
| 97 | \( 1 + (4.57 - 2.64i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.97148944175040621467503891343, −10.23200053712602902082453944329, −9.782725348308275647848677338898, −8.130010163219209426760282736536, −7.05933550616352083521633288880, −6.68972147865585845347424688424, −5.23671814141217563271449576986, −4.42431954474027780487299996079, −2.72484386478169911200018434742, −1.76585263281501333864002816753,
0.35816564600247121093320787677, 2.62788002944407469267251501398, 4.54998619837860617454791118050, 5.24282156595012684028780979261, 5.70421315741467803510777677259, 6.73826495424014224963268069107, 8.282301787841240186512629718435, 8.741161264522920130885243995878, 9.687288876385305221747497719706, 10.50792071174222785293320892561