| L(s) = 1 | + (1.47 − 1.35i)2-s + (0.350 − 3.98i)4-s − 10.9·7-s + (−4.86 − 6.35i)8-s + 11.3i·11-s − 10.8i·13-s + (−16.1 + 14.7i)14-s + (−15.7 − 2.79i)16-s + 15.8i·17-s + 24.9i·19-s + (15.3 + 16.7i)22-s + 20.9·23-s + (−14.5 − 15.9i)26-s + (−3.83 + 43.5i)28-s + 22.8·29-s + ⋯ |
| L(s) = 1 | + (0.737 − 0.675i)2-s + (0.0876 − 0.996i)4-s − 1.55·7-s + (−0.608 − 0.793i)8-s + 1.03i·11-s − 0.831i·13-s + (−1.15 + 1.05i)14-s + (−0.984 − 0.174i)16-s + 0.929i·17-s + 1.31i·19-s + (0.697 + 0.761i)22-s + 0.911·23-s + (−0.561 − 0.613i)26-s + (−0.136 + 1.55i)28-s + 0.786·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.523 - 0.851i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.523 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.118393864\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.118393864\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.47 + 1.35i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 10.9T + 49T^{2} \) |
| 11 | \( 1 - 11.3iT - 121T^{2} \) |
| 13 | \( 1 + 10.8iT - 169T^{2} \) |
| 17 | \( 1 - 15.8iT - 289T^{2} \) |
| 19 | \( 1 - 24.9iT - 361T^{2} \) |
| 23 | \( 1 - 20.9T + 529T^{2} \) |
| 29 | \( 1 - 22.8T + 841T^{2} \) |
| 31 | \( 1 - 22.7iT - 961T^{2} \) |
| 37 | \( 1 - 19.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 17T + 1.68e3T^{2} \) |
| 43 | \( 1 - 6.51T + 1.84e3T^{2} \) |
| 47 | \( 1 + 38.9T + 2.20e3T^{2} \) |
| 53 | \( 1 + 13.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 95.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 92.0T + 3.72e3T^{2} \) |
| 67 | \( 1 + 54.1T + 4.48e3T^{2} \) |
| 71 | \( 1 - 68.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 44.1iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 81.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 27.9T + 6.88e3T^{2} \) |
| 89 | \( 1 + 42.1T + 7.92e3T^{2} \) |
| 97 | \( 1 + 154. iT - 9.40e3T^{2} \) |
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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
−10.18193261130077473523709304878, −9.611682852242216584796607313310, −8.552178327804076935334808733158, −7.23125285475101234218006836972, −6.39677126655216022938373909595, −5.69175426285992398511123629965, −4.56808757559415975964261627052, −3.52904525244434184801037809797, −2.82703933087186834683200309175, −1.40067204441139573851562845954,
0.28055180028908727482558445510, 2.71975015559850094154722482743, 3.32833250701421017528218749306, 4.48196959376625998711247049462, 5.46460602419834055155998351772, 6.57330112726060857803841905046, 6.75118995160611394021766236996, 7.912771404498919746031913830364, 9.148372156173469978658604088565, 9.305150258592561068542885898906
