L(s) = 1 | − i·2-s + (1.51 + 0.835i)3-s − 4-s + (−0.5 − 0.866i)5-s + (0.835 − 1.51i)6-s + (2.64 + 0.162i)7-s + i·8-s + (1.60 + 2.53i)9-s + (−0.866 + 0.5i)10-s + (−0.650 − 0.375i)11-s + (−1.51 − 0.835i)12-s + (2.72 + 1.57i)13-s + (0.162 − 2.64i)14-s + (−0.0350 − 1.73i)15-s + 16-s + (−0.433 − 0.750i)17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (0.875 + 0.482i)3-s − 0.5·4-s + (−0.223 − 0.387i)5-s + (0.341 − 0.619i)6-s + (0.998 + 0.0612i)7-s + 0.353i·8-s + (0.534 + 0.845i)9-s + (−0.273 + 0.158i)10-s + (−0.196 − 0.113i)11-s + (−0.437 − 0.241i)12-s + (0.754 + 0.435i)13-s + (0.0433 − 0.705i)14-s + (−0.00906 − 0.447i)15-s + 0.250·16-s + (−0.105 − 0.181i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.879 + 0.476i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.879 + 0.476i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.00404 - 0.508094i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.00404 - 0.508094i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.51 - 0.835i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (-2.64 - 0.162i)T \) |
good | 11 | \( 1 + (0.650 + 0.375i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.72 - 1.57i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (0.433 + 0.750i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.23 - 0.713i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.38 + 2.53i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (5.23 - 3.02i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 6.21iT - 31T^{2} \) |
| 37 | \( 1 + (-3.60 + 6.24i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.08 + 3.61i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.69 - 6.40i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 3.45T + 47T^{2} \) |
| 53 | \( 1 + (-0.790 + 0.456i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 8.43T + 59T^{2} \) |
| 61 | \( 1 - 10.7iT - 61T^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 + 5.43iT - 71T^{2} \) |
| 73 | \( 1 + (-0.539 + 0.311i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 - 0.00585T + 79T^{2} \) |
| 83 | \( 1 + (1.01 + 1.76i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.22 + 2.12i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (9.23 - 5.33i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.79566227824426069867444340306, −9.450729719135608113754965019418, −8.972981911075646381681482976288, −8.133867940527420781583047679256, −7.39361783613270393695010212910, −5.64360906152233060668068619515, −4.60959816154095756875958127586, −3.91737200283510735521110161904, −2.66674522722353249703267442113, −1.45532418289624581661529200543,
1.39972455419111218896668610647, 2.97160993129776856699970216028, 4.06496259987245837946743216121, 5.23363005137765733925478776092, 6.39468324513770242643961168541, 7.34275930191820369834771796709, 7.933407054171278801636446545779, 8.645017299354882896703099349350, 9.503733617378186934632276903496, 10.64392294132513511871773215366