L(s) = 1 | + (0.173 − 0.984i)2-s + (−0.0864 − 0.0725i)3-s + (−0.939 − 0.342i)4-s + (−0.0864 + 0.0725i)6-s + (−0.772 − 1.33i)7-s + (−0.5 + 0.866i)8-s + (−0.518 − 2.94i)9-s + (0.653 − 1.13i)11-s + (0.0564 + 0.0977i)12-s + (0.437 − 0.367i)13-s + (−1.45 + 0.528i)14-s + (0.766 + 0.642i)16-s + (−0.878 + 4.98i)17-s − 2.98·18-s + (−0.546 − 4.32i)19-s + ⋯ |
L(s) = 1 | + (0.122 − 0.696i)2-s + (−0.0499 − 0.0418i)3-s + (−0.469 − 0.171i)4-s + (−0.0353 + 0.0296i)6-s + (−0.291 − 0.505i)7-s + (−0.176 + 0.306i)8-s + (−0.172 − 0.980i)9-s + (0.196 − 0.341i)11-s + (0.0162 + 0.0282i)12-s + (0.121 − 0.101i)13-s + (−0.387 + 0.141i)14-s + (0.191 + 0.160i)16-s + (−0.212 + 1.20i)17-s − 0.704·18-s + (−0.125 − 0.992i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0646i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0280689 + 0.866881i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0280689 + 0.866881i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.173 + 0.984i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (0.546 + 4.32i)T \) |
good | 3 | \( 1 + (0.0864 + 0.0725i)T + (0.520 + 2.95i)T^{2} \) |
| 7 | \( 1 + (0.772 + 1.33i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.653 + 1.13i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.437 + 0.367i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.878 - 4.98i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (4.86 + 1.77i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.660 - 3.74i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (1.49 + 2.58i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 3.31T + 37T^{2} \) |
| 41 | \( 1 + (8.57 + 7.19i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (8.85 - 3.22i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (0.368 + 2.08i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (7.19 + 2.61i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (0.0463 - 0.262i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-3.05 - 1.11i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (1.57 + 8.94i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-15.2 + 5.55i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (2.14 + 1.80i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-3.19 - 2.68i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-4.63 - 8.03i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (4.42 - 3.71i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (0.500 - 2.83i)T + (-91.1 - 33.1i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.756351469395601670806095380990, −8.862473859682153601095688631965, −8.216614558777538263931735561505, −6.83835104918606751431472927792, −6.25144245962541601598896043110, −5.10119267108084717007782194332, −3.89455988961314902102453398860, −3.35260363939921756883465962125, −1.86323160079845992221625632130, −0.37849328032389022972868291267,
1.96388746206689266718906000461, 3.28223517836354316199695675047, 4.51873766840248590208686943223, 5.31829086249061033958201782338, 6.15778927313165249470195919110, 7.07338431034392659592481197941, 7.952540726733756136993048672429, 8.607438144333366503170542936297, 9.642058875146193505443334291740, 10.19576779305194283131815987685