| L(s) = 1 | + (−0.912 − 4.58i)3-s + (7.13 + 4.76i)5-s + (7.38 − 4.93i)7-s + (−11.8 + 4.92i)9-s + (−10.3 − 2.06i)11-s + (7.09 − 7.09i)13-s + (15.3 − 37.0i)15-s + (−9.06 − 14.3i)17-s + (0.265 + 0.109i)19-s + (−29.3 − 29.3i)21-s + (−7.43 + 37.3i)23-s + (18.5 + 44.8i)25-s + (10.0 + 15.0i)27-s + (11.2 − 16.8i)29-s + (13.1 − 2.62i)31-s + ⋯ |
| L(s) = 1 | + (−0.304 − 1.52i)3-s + (1.42 + 0.952i)5-s + (1.05 − 0.704i)7-s + (−1.32 + 0.546i)9-s + (−0.943 − 0.187i)11-s + (0.545 − 0.545i)13-s + (1.02 − 2.46i)15-s + (−0.533 − 0.845i)17-s + (0.0139 + 0.00578i)19-s + (−1.39 − 1.39i)21-s + (−0.323 + 1.62i)23-s + (0.743 + 1.79i)25-s + (0.371 + 0.556i)27-s + (0.387 − 0.579i)29-s + (0.425 − 0.0845i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.304 + 0.952i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.304 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.32857 - 0.969881i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.32857 - 0.969881i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + (9.06 + 14.3i)T \) |
| good | 3 | \( 1 + (0.912 + 4.58i)T + (-8.31 + 3.44i)T^{2} \) |
| 5 | \( 1 + (-7.13 - 4.76i)T + (9.56 + 23.0i)T^{2} \) |
| 7 | \( 1 + (-7.38 + 4.93i)T + (18.7 - 45.2i)T^{2} \) |
| 11 | \( 1 + (10.3 + 2.06i)T + (111. + 46.3i)T^{2} \) |
| 13 | \( 1 + (-7.09 + 7.09i)T - 169iT^{2} \) |
| 19 | \( 1 + (-0.265 - 0.109i)T + (255. + 255. i)T^{2} \) |
| 23 | \( 1 + (7.43 - 37.3i)T + (-488. - 202. i)T^{2} \) |
| 29 | \( 1 + (-11.2 + 16.8i)T + (-321. - 776. i)T^{2} \) |
| 31 | \( 1 + (-13.1 + 2.62i)T + (887. - 367. i)T^{2} \) |
| 37 | \( 1 + (-8.07 - 40.5i)T + (-1.26e3 + 523. i)T^{2} \) |
| 41 | \( 1 + (41.2 - 27.5i)T + (643. - 1.55e3i)T^{2} \) |
| 43 | \( 1 + (-33.4 + 13.8i)T + (1.30e3 - 1.30e3i)T^{2} \) |
| 47 | \( 1 + (28.3 - 28.3i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-77.8 - 32.2i)T + (1.98e3 + 1.98e3i)T^{2} \) |
| 59 | \( 1 + (27.7 + 66.9i)T + (-2.46e3 + 2.46e3i)T^{2} \) |
| 61 | \( 1 + (26.1 + 39.1i)T + (-1.42e3 + 3.43e3i)T^{2} \) |
| 67 | \( 1 - 46.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (-24.0 - 120. i)T + (-4.65e3 + 1.92e3i)T^{2} \) |
| 73 | \( 1 + (31.3 + 20.9i)T + (2.03e3 + 4.92e3i)T^{2} \) |
| 79 | \( 1 + (44.2 + 8.81i)T + (5.76e3 + 2.38e3i)T^{2} \) |
| 83 | \( 1 + (16.6 - 40.1i)T + (-4.87e3 - 4.87e3i)T^{2} \) |
| 89 | \( 1 + (-74.9 - 74.9i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + (-78.3 + 117. i)T + (-3.60e3 - 8.69e3i)T^{2} \) |
| 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
−13.28687302150803383676669107788, −11.65226184328510095238426276758, −10.89991852846031490461048264648, −9.884087725211062222150473626287, −8.118808419301120790815670174314, −7.29967538390782096867486400273, −6.29564091062056567962582682829, −5.31503890555548264026004983069, −2.65286054505093219353008259748, −1.38845713581503641879751524715,
2.11180302991466409391181275217, 4.44683041438246546653436643398, 5.16991262383398525921522423238, 6.05159680048192639790944360750, 8.540133444077129274964433899202, 8.974081455994180592700789091351, 10.21685047899414803273294689086, 10.75389172093383923628509244916, 12.10692311059960147928690146045, 13.21897618294156939245035821926
