| L(s) = 1 | − 1.41i·2-s − 2.00·4-s + 2.34i·5-s + 9.61·7-s + 2.82i·8-s + 3.32·10-s − 5.69·13-s − 13.5i·14-s + 4.00·16-s − 14.8i·17-s − 30.9·19-s − 4.69i·20-s + 5.53i·23-s + 19.4·25-s + 8.05i·26-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s + 0.469i·5-s + 1.37·7-s + 0.353i·8-s + 0.332·10-s − 0.438·13-s − 0.970i·14-s + 0.250·16-s − 0.872i·17-s − 1.62·19-s − 0.234i·20-s + 0.240i·23-s + 0.779·25-s + 0.309i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.308417599\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.308417599\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 2.34iT - 25T^{2} \) |
| 7 | \( 1 - 9.61T + 49T^{2} \) |
| 13 | \( 1 + 5.69T + 169T^{2} \) |
| 17 | \( 1 + 14.8iT - 289T^{2} \) |
| 19 | \( 1 + 30.9T + 361T^{2} \) |
| 23 | \( 1 - 5.53iT - 529T^{2} \) |
| 29 | \( 1 + 21.4iT - 841T^{2} \) |
| 31 | \( 1 + 36.8T + 961T^{2} \) |
| 37 | \( 1 - 4.13T + 1.36e3T^{2} \) |
| 41 | \( 1 + 64.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 28.5T + 1.84e3T^{2} \) |
| 47 | \( 1 - 9.03iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 46.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 59.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 1.97T + 3.72e3T^{2} \) |
| 67 | \( 1 + 100.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 90.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 108.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 130.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 85.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 131. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 19.5T + 9.40e3T^{2} \) |
| 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
−8.704292351296988298099037656215, −7.84403556449865450927811136673, −7.20391974424531268618603767442, −6.15502794683655989988828848819, −5.08001111319046600520565994239, −4.56154828800309144507821465282, −3.55617630228111463410469207593, −2.41248301862170102849959083798, −1.76523715861752404651206804792, −0.32607510684814639697377331835,
1.22683815643282226755833910459, 2.20727215587496326558848100168, 3.74786876494502548086347884815, 4.66937528224514974650361766565, 5.07317680718912639174743154412, 6.08003464161205339927602031580, 6.88447185148960445832767126832, 7.76660916731794172446661943041, 8.470499558186314594385305450973, 8.759024394212500172603566122022
