| L(s) = 1 | − 4.44i·3-s − 4.97·5-s − 12.2i·7-s − 10.7·9-s − 13.4i·11-s − 14.1·13-s + 22.1i·15-s − 5.89·17-s + 4.35i·19-s − 54.5·21-s − 0.906i·23-s − 0.202·25-s + 7.99i·27-s + 10.3·29-s − 43.2i·31-s + ⋯ |
| L(s) = 1 | − 1.48i·3-s − 0.995·5-s − 1.75i·7-s − 1.19·9-s − 1.22i·11-s − 1.09·13-s + 1.47i·15-s − 0.346·17-s + 0.229i·19-s − 2.59·21-s − 0.0394i·23-s − 0.00809·25-s + 0.296i·27-s + 0.358·29-s − 1.39i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8317173031\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8317173031\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - 4.35iT \) |
| good | 3 | \( 1 + 4.44iT - 9T^{2} \) |
| 5 | \( 1 + 4.97T + 25T^{2} \) |
| 7 | \( 1 + 12.2iT - 49T^{2} \) |
| 11 | \( 1 + 13.4iT - 121T^{2} \) |
| 13 | \( 1 + 14.1T + 169T^{2} \) |
| 17 | \( 1 + 5.89T + 289T^{2} \) |
| 23 | \( 1 + 0.906iT - 529T^{2} \) |
| 29 | \( 1 - 10.3T + 841T^{2} \) |
| 31 | \( 1 + 43.2iT - 961T^{2} \) |
| 37 | \( 1 - 1.61T + 1.36e3T^{2} \) |
| 41 | \( 1 - 69.3T + 1.68e3T^{2} \) |
| 43 | \( 1 + 32.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 38.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 8.31T + 2.80e3T^{2} \) |
| 59 | \( 1 - 20.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 118.T + 3.72e3T^{2} \) |
| 67 | \( 1 - 57.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 11.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 23.5T + 5.32e3T^{2} \) |
| 79 | \( 1 - 0.286iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 24.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 43.7T + 7.92e3T^{2} \) |
| 97 | \( 1 - 115.T + 9.40e3T^{2} \) |
| 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
−8.519709371169728420024453217773, −7.80981127327547721364430600338, −7.38088570391028220794986140426, −6.77043334602033981322070723084, −5.78392656041377157973917869631, −4.37630640631414518308312521246, −3.64351712420818485341005801852, −2.37130795256991998917069410889, −0.868932950164980374274077533293, −0.32279116782426924741832268510,
2.26858560813791960537583086911, 3.15100884981583976192672899879, 4.35915227569055673900954522300, 4.82235933858318503787865784154, 5.63460600486444931734939058497, 6.89121716139142731796278405825, 7.88640124062461211500374062533, 8.751528031894245843642587125035, 9.433407403516190490447432736005, 9.909708796044777913189789183546
