L(s) = 1 | − i·2-s + (0.866 − 1.5i)3-s − 4-s + (−0.5 + 0.866i)5-s + (−1.5 − 0.866i)6-s + (−2.5 + 0.866i)7-s + i·8-s + (−1.5 − 2.59i)9-s + (0.866 + 0.5i)10-s + (−4.09 + 2.36i)11-s + (−0.866 + 1.5i)12-s + (−3 + 1.73i)13-s + (0.866 + 2.5i)14-s + (0.866 + 1.5i)15-s + 16-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (0.499 − 0.866i)3-s − 0.5·4-s + (−0.223 + 0.387i)5-s + (−0.612 − 0.353i)6-s + (−0.944 + 0.327i)7-s + 0.353i·8-s + (−0.5 − 0.866i)9-s + (0.273 + 0.158i)10-s + (−1.23 + 0.713i)11-s + (−0.249 + 0.433i)12-s + (−0.832 + 0.480i)13-s + (0.231 + 0.668i)14-s + (0.223 + 0.387i)15-s + 0.250·16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.281 - 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.281 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.866 + 1.5i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (2.5 - 0.866i)T \) |
good | 11 | \( 1 + (4.09 - 2.36i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (3 - 1.73i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.09 - 0.633i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.19 - 1.26i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (5.59 + 3.23i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 8.19iT - 31T^{2} \) |
| 37 | \( 1 + (3.09 + 5.36i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.5 - 7.79i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.59 - 2.76i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 9T + 47T^{2} \) |
| 53 | \( 1 + (-6.29 - 3.63i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 2.19T + 59T^{2} \) |
| 61 | \( 1 + 12.9iT - 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 4.73iT - 71T^{2} \) |
| 73 | \( 1 + (-6 - 3.46i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 14.5T + 79T^{2} \) |
| 83 | \( 1 + (5.59 - 9.69i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-2.19 - 3.80i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (7.39 + 4.26i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.705904262587716238925016345933, −9.479237533600101899617652561490, −8.123947896503976530483251752122, −7.45580786820551258123825439594, −6.53037265933990513408761694785, −5.39415029064606927896043746479, −3.96578539162292035218827257301, −2.79502513973306058405091944143, −2.14615554663578379021417444230, 0,
2.82510419271929226058787298081, 3.69831776505631669409499329688, 4.96785759406228997175136856123, 5.51329322145192892682867486972, 6.90127023460504338083326593194, 7.79517787228108252929192716538, 8.618767238160573727136728067158, 9.318098855297132481919530786948, 10.31579919244191883308378181727