| L(s) = 1 | + (−2.20 − 4.82i)3-s + (−5.92 + 5.13i)5-s + (3.26 − 5.07i)7-s + (−12.4 + 14.4i)9-s + (−19.6 + 2.82i)11-s + (4.36 − 2.80i)13-s + (37.7 + 17.2i)15-s + (−5.72 − 19.4i)17-s + (2.95 − 10.0i)19-s + (−31.6 − 4.55i)21-s + (8.46 − 21.3i)23-s + (5.18 − 36.0i)25-s + (51.2 + 15.0i)27-s + (4.07 − 1.19i)29-s + (−3.96 + 8.67i)31-s + ⋯ |
| L(s) = 1 | + (−0.733 − 1.60i)3-s + (−1.18 + 1.02i)5-s + (0.466 − 0.725i)7-s + (−1.38 + 1.60i)9-s + (−1.78 + 0.256i)11-s + (0.336 − 0.215i)13-s + (2.51 + 1.14i)15-s + (−0.336 − 1.14i)17-s + (0.155 − 0.529i)19-s + (−1.50 − 0.216i)21-s + (0.367 − 0.929i)23-s + (0.207 − 1.44i)25-s + (1.89 + 0.557i)27-s + (0.140 − 0.0412i)29-s + (−0.127 + 0.279i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.104i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.994 - 0.104i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0191119 + 0.364091i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0191119 + 0.364091i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 + (-8.46 + 21.3i)T \) |
| good | 3 | \( 1 + (2.20 + 4.82i)T + (-5.89 + 6.80i)T^{2} \) |
| 5 | \( 1 + (5.92 - 5.13i)T + (3.55 - 24.7i)T^{2} \) |
| 7 | \( 1 + (-3.26 + 5.07i)T + (-20.3 - 44.5i)T^{2} \) |
| 11 | \( 1 + (19.6 - 2.82i)T + (116. - 34.0i)T^{2} \) |
| 13 | \( 1 + (-4.36 + 2.80i)T + (70.2 - 153. i)T^{2} \) |
| 17 | \( 1 + (5.72 + 19.4i)T + (-243. + 156. i)T^{2} \) |
| 19 | \( 1 + (-2.95 + 10.0i)T + (-303. - 195. i)T^{2} \) |
| 29 | \( 1 + (-4.07 + 1.19i)T + (707. - 454. i)T^{2} \) |
| 31 | \( 1 + (3.96 - 8.67i)T + (-629. - 726. i)T^{2} \) |
| 37 | \( 1 + (19.3 + 16.7i)T + (194. + 1.35e3i)T^{2} \) |
| 41 | \( 1 + (-4.29 - 4.95i)T + (-239. + 1.66e3i)T^{2} \) |
| 43 | \( 1 + (20.6 - 9.45i)T + (1.21e3 - 1.39e3i)T^{2} \) |
| 47 | \( 1 + 20.1T + 2.20e3T^{2} \) |
| 53 | \( 1 + (24.5 - 38.2i)T + (-1.16e3 - 2.55e3i)T^{2} \) |
| 59 | \( 1 + (-67.0 + 43.0i)T + (1.44e3 - 3.16e3i)T^{2} \) |
| 61 | \( 1 + (11.4 + 5.21i)T + (2.43e3 + 2.81e3i)T^{2} \) |
| 67 | \( 1 + (85.8 + 12.3i)T + (4.30e3 + 1.26e3i)T^{2} \) |
| 71 | \( 1 + (-14.8 + 103. i)T + (-4.83e3 - 1.42e3i)T^{2} \) |
| 73 | \( 1 + (-103. - 30.4i)T + (4.48e3 + 2.88e3i)T^{2} \) |
| 79 | \( 1 + (11.2 + 17.5i)T + (-2.59e3 + 5.67e3i)T^{2} \) |
| 83 | \( 1 + (-74.8 - 64.8i)T + (980. + 6.81e3i)T^{2} \) |
| 89 | \( 1 + (66.1 - 30.2i)T + (5.18e3 - 5.98e3i)T^{2} \) |
| 97 | \( 1 + (-2.46 + 2.13i)T + (1.33e3 - 9.31e3i)T^{2} \) |
| 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
−13.18516062885588253104622001972, −12.17766420712293711572497417942, −11.10711659847245672008234229176, −10.70834634227797289742472758726, −8.067788011155312956612747388787, −7.45162032952065894300162974457, −6.72117815157326368480970794298, −4.99526086296460720583914122714, −2.72265762500097759859985970895, −0.30276942569117225665687052391,
3.69237625178521544609170257523, 4.89064381156412307337077909756, 5.60150146133040005826743217164, 8.076277740993497165766511180334, 8.817784425454041025910991251475, 10.20654559581311128229714995502, 11.17203352035428822188953550133, 11.92610070884577307000287891228, 13.05444989224859626312371799599, 14.98734426602519799749387550983
