L(s) = 1 | − 2-s + (0.841 + 1.51i)3-s + 4-s + (2.84 − 1.64i)5-s + (−0.841 − 1.51i)6-s + (1.87 + 1.86i)7-s − 8-s + (−1.58 + 2.54i)9-s + (−2.84 + 1.64i)10-s + (−1.03 − 1.79i)11-s + (0.841 + 1.51i)12-s + (3.53 + 0.716i)13-s + (−1.87 − 1.86i)14-s + (4.88 + 2.92i)15-s + 16-s + 1.12·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.486 + 0.873i)3-s + 0.5·4-s + (1.27 − 0.735i)5-s + (−0.343 − 0.617i)6-s + (0.707 + 0.706i)7-s − 0.353·8-s + (−0.527 + 0.849i)9-s + (−0.900 + 0.519i)10-s + (−0.312 − 0.540i)11-s + (0.243 + 0.436i)12-s + (0.980 + 0.198i)13-s + (−0.500 − 0.499i)14-s + (1.26 + 0.755i)15-s + 0.250·16-s + 0.272·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.758 - 0.651i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.758 - 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.52344 + 0.564393i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.52344 + 0.564393i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (-0.841 - 1.51i)T \) |
| 7 | \( 1 + (-1.87 - 1.86i)T \) |
| 13 | \( 1 + (-3.53 - 0.716i)T \) |
good | 5 | \( 1 + (-2.84 + 1.64i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.03 + 1.79i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 - 1.12T + 17T^{2} \) |
| 19 | \( 1 + (0.505 - 0.875i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 3.50iT - 23T^{2} \) |
| 29 | \( 1 + (7.97 + 4.60i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1.86 - 3.23i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 6.97iT - 37T^{2} \) |
| 41 | \( 1 + (-8.52 - 4.92i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.35 - 5.81i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.05 + 0.609i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (5.05 + 2.92i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 9.80iT - 59T^{2} \) |
| 61 | \( 1 + (0.209 + 0.120i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (10.2 - 5.90i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.94 - 6.83i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.878 + 1.52i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.48 + 4.31i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 0.999iT - 83T^{2} \) |
| 89 | \( 1 + 7.61iT - 89T^{2} \) |
| 97 | \( 1 + (3.21 + 5.57i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.78076048630651407896120794716, −9.771514934297169301794168985410, −9.094182540657047972920867012821, −8.609880453995901728179997492304, −7.76816956493017183970701554774, −5.92908453610225991079813327526, −5.57903050683342800277719898818, −4.28407894145372736406602427025, −2.70280248648935893099008695580, −1.62595666343983434029003753088,
1.38900311083116223027143907340, 2.24791726365631091470158124774, 3.55569974770693618326369678968, 5.49383632533697184859611057847, 6.37475438312587610137841645753, 7.28484627733128831736794270694, 7.86156167523575245476568945968, 9.004213145835633946057513491606, 9.707663165003313077986930454904, 10.72367908123948205155921363488