L(s) = 1 | + (0.5 + 0.866i)2-s + (−1.63 − 0.559i)3-s + (−0.499 + 0.866i)4-s + 2.94i·5-s + (−0.334 − 1.69i)6-s + (−1.08 + 2.41i)7-s − 0.999·8-s + (2.37 + 1.83i)9-s + (−2.55 + 1.47i)10-s + (−1.48 − 2.57i)11-s + (1.30 − 1.13i)12-s + (−2.33 − 2.75i)13-s + (−2.63 + 0.272i)14-s + (1.65 − 4.83i)15-s + (−0.5 − 0.866i)16-s + (2.27 − 3.93i)17-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.946 − 0.323i)3-s + (−0.249 + 0.433i)4-s + 1.31i·5-s + (−0.136 − 0.693i)6-s + (−0.408 + 0.912i)7-s − 0.353·8-s + (0.791 + 0.611i)9-s + (−0.807 + 0.466i)10-s + (−0.448 − 0.776i)11-s + (0.376 − 0.329i)12-s + (−0.646 − 0.762i)13-s + (−0.703 + 0.0727i)14-s + (0.426 − 1.24i)15-s + (−0.125 − 0.216i)16-s + (0.551 − 0.955i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.115458 - 0.488315i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.115458 - 0.488315i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (1.63 + 0.559i)T \) |
| 7 | \( 1 + (1.08 - 2.41i)T \) |
| 13 | \( 1 + (2.33 + 2.75i)T \) |
good | 5 | \( 1 - 2.94iT - 5T^{2} \) |
| 11 | \( 1 + (1.48 + 2.57i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-2.27 + 3.93i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.30 - 5.71i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.82 - 1.05i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.262 + 0.151i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 7.31T + 31T^{2} \) |
| 37 | \( 1 + (5.19 - 2.99i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.22 + 2.44i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4.53 - 7.84i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 7.77iT - 47T^{2} \) |
| 53 | \( 1 - 1.95iT - 53T^{2} \) |
| 59 | \( 1 + (-5.73 - 3.30i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.23 - 3.59i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.91 - 3.99i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (4.69 - 8.13i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 9.10T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 - 5.68iT - 83T^{2} \) |
| 89 | \( 1 + (12.8 - 7.44i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (5.31 - 9.20i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.39693061320546719688503481455, −10.47329260673240254348983435922, −9.795732726750978575041361001826, −8.316525863647303812741840710930, −7.44911788491408025441063092322, −6.65848120756874672569310401062, −5.77450429057403242725330781666, −5.30647054733941663731310557504, −3.57358533972191618797302895585, −2.50203611462574290236021549492,
0.28013433777394687303762474240, 1.78210082947880392620880745796, 3.85672155095048566988998397493, 4.59718324988433970690415740190, 5.23482236243533835596362215269, 6.46955986233986287236960918199, 7.44106634311258623749942983390, 8.884749436536963272342497132061, 9.647832413306494150932588161055, 10.39753987999950149135726756920