| L(s) = 1 | + (−2.05 + 1.49i)2-s + (−2.06 + 2.17i)3-s + (0.756 − 2.32i)4-s + (0.996 − 7.55i)6-s − 6.97i·7-s + (−1.21 − 3.74i)8-s + (−0.465 − 8.98i)9-s + (7.98 + 10.9i)11-s + (3.50 + 6.45i)12-s + (−0.259 + 0.357i)13-s + (10.4 + 14.3i)14-s + (16.0 + 11.6i)16-s + (7.94 + 24.4i)17-s + (14.3 + 17.7i)18-s + (−3.70 − 11.4i)19-s + ⋯ |
| L(s) = 1 | + (−1.02 + 0.746i)2-s + (−0.688 + 0.725i)3-s + (0.189 − 0.581i)4-s + (0.166 − 1.25i)6-s − 0.996i·7-s + (−0.152 − 0.468i)8-s + (−0.0517 − 0.998i)9-s + (0.725 + 0.999i)11-s + (0.291 + 0.537i)12-s + (−0.0199 + 0.0275i)13-s + (0.743 + 1.02i)14-s + (1.00 + 0.727i)16-s + (0.467 + 1.43i)17-s + (0.798 + 0.987i)18-s + (−0.195 − 0.600i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.699 - 0.714i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.699 - 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.244190 + 0.580931i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.244190 + 0.580931i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.06 - 2.17i)T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + (2.05 - 1.49i)T + (1.23 - 3.80i)T^{2} \) |
| 7 | \( 1 + 6.97iT - 49T^{2} \) |
| 11 | \( 1 + (-7.98 - 10.9i)T + (-37.3 + 115. i)T^{2} \) |
| 13 | \( 1 + (0.259 - 0.357i)T + (-52.2 - 160. i)T^{2} \) |
| 17 | \( 1 + (-7.94 - 24.4i)T + (-233. + 169. i)T^{2} \) |
| 19 | \( 1 + (3.70 + 11.4i)T + (-292. + 212. i)T^{2} \) |
| 23 | \( 1 + (-19.1 + 13.9i)T + (163. - 503. i)T^{2} \) |
| 29 | \( 1 + (32.2 + 10.4i)T + (680. + 494. i)T^{2} \) |
| 31 | \( 1 + (-7.00 - 21.5i)T + (-777. + 564. i)T^{2} \) |
| 37 | \( 1 + (-24.1 + 33.2i)T + (-423. - 1.30e3i)T^{2} \) |
| 41 | \( 1 + (11.1 - 15.3i)T + (-519. - 1.59e3i)T^{2} \) |
| 43 | \( 1 - 40.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (7.83 - 24.1i)T + (-1.78e3 - 1.29e3i)T^{2} \) |
| 53 | \( 1 + (22.0 - 67.9i)T + (-2.27e3 - 1.65e3i)T^{2} \) |
| 59 | \( 1 + (16.0 - 22.0i)T + (-1.07e3 - 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-77.8 + 56.5i)T + (1.14e3 - 3.53e3i)T^{2} \) |
| 67 | \( 1 + (92.0 - 29.9i)T + (3.63e3 - 2.63e3i)T^{2} \) |
| 71 | \( 1 + (-15.2 - 4.94i)T + (4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (-23.6 - 32.5i)T + (-1.64e3 + 5.06e3i)T^{2} \) |
| 79 | \( 1 + (3.55 - 10.9i)T + (-5.04e3 - 3.66e3i)T^{2} \) |
| 83 | \( 1 + (9.61 + 29.5i)T + (-5.57e3 + 4.04e3i)T^{2} \) |
| 89 | \( 1 + (-35.6 - 49.0i)T + (-2.44e3 + 7.53e3i)T^{2} \) |
| 97 | \( 1 + (-125. - 40.9i)T + (7.61e3 + 5.53e3i)T^{2} \) |
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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
−11.07555988376155698047330273865, −10.38182549808707198062590944525, −9.618159443355147470450312073219, −8.891681665349526520139672114831, −7.68599851983204752300299830811, −6.84829432020974595901901406944, −6.08266872764247023563389618534, −4.55870894707495478085312218547, −3.71925654071413993466178307012, −1.06462513492513132782896290895,
0.55721730822219025839945705726, 1.82711898825621456356067551672, 3.08684592687407065134259013728, 5.24470283669308803622759608737, 5.91546188545070218179630691839, 7.20272359642822593300130944683, 8.270146945027483786101110433735, 9.046195709970331903680676645816, 9.880816248139254884068411334582, 11.04355768576456018472088784707
