| L(s) = 1 | + (−1.90 − 3.16i)2-s + (2.89 − 0.777i)3-s + (−4.50 + 8.50i)4-s + (−0.853 − 7.84i)5-s + (−7.97 − 7.68i)6-s + (12.1 − 4.09i)7-s + (20.7 − 1.12i)8-s + (7.79 − 4.50i)9-s + (−23.1 + 17.6i)10-s + (2.40 + 1.62i)11-s + (−6.45 + 28.1i)12-s + (−1.30 − 1.23i)13-s + (−36.0 − 30.6i)14-s + (−8.57 − 22.0i)15-s + (−21.4 − 31.6i)16-s + (−5.27 + 15.6i)17-s + ⋯ |
| L(s) = 1 | + (−0.951 − 1.58i)2-s + (0.965 − 0.259i)3-s + (−1.12 + 2.12i)4-s + (−0.170 − 1.56i)5-s + (−1.32 − 1.28i)6-s + (1.73 − 0.584i)7-s + (2.59 − 0.140i)8-s + (0.865 − 0.500i)9-s + (−2.31 + 1.76i)10-s + (0.218 + 0.148i)11-s + (−0.537 + 2.34i)12-s + (−0.100 − 0.0949i)13-s + (−2.57 − 2.18i)14-s + (−0.571 − 1.47i)15-s + (−1.33 − 1.97i)16-s + (−0.310 + 0.920i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.144i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.989 + 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0975270 - 1.34226i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0975270 - 1.34226i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.89 + 0.777i)T \) |
| 59 | \( 1 + (12.2 + 57.7i)T \) |
| good | 2 | \( 1 + (1.90 + 3.16i)T + (-1.87 + 3.53i)T^{2} \) |
| 5 | \( 1 + (0.853 + 7.84i)T + (-24.4 + 5.37i)T^{2} \) |
| 7 | \( 1 + (-12.1 + 4.09i)T + (39.0 - 29.6i)T^{2} \) |
| 11 | \( 1 + (-2.40 - 1.62i)T + (44.7 + 112. i)T^{2} \) |
| 13 | \( 1 + (1.30 + 1.23i)T + (9.14 + 168. i)T^{2} \) |
| 17 | \( 1 + (5.27 - 15.6i)T + (-230. - 174. i)T^{2} \) |
| 19 | \( 1 + (0.0242 + 0.148i)T + (-342. + 115. i)T^{2} \) |
| 23 | \( 1 + (30.3 - 8.42i)T + (453. - 272. i)T^{2} \) |
| 29 | \( 1 + (-1.00 + 1.66i)T + (-393. - 743. i)T^{2} \) |
| 31 | \( 1 + (6.01 - 36.6i)T + (-910. - 306. i)T^{2} \) |
| 37 | \( 1 + (1.35 - 24.9i)T + (-1.36e3 - 148. i)T^{2} \) |
| 41 | \( 1 + (-64.5 - 17.9i)T + (1.44e3 + 866. i)T^{2} \) |
| 43 | \( 1 + (19.8 + 29.2i)T + (-684. + 1.71e3i)T^{2} \) |
| 47 | \( 1 + (4.02 - 37.0i)T + (-2.15e3 - 474. i)T^{2} \) |
| 53 | \( 1 + (18.5 - 24.4i)T + (-751. - 2.70e3i)T^{2} \) |
| 61 | \( 1 + (-39.1 + 23.5i)T + (1.74e3 - 3.28e3i)T^{2} \) |
| 67 | \( 1 + (2.79 + 51.5i)T + (-4.46e3 + 485. i)T^{2} \) |
| 71 | \( 1 + (2.48 - 22.8i)T + (-4.92e3 - 1.08e3i)T^{2} \) |
| 73 | \( 1 + (20.7 - 24.3i)T + (-862. - 5.25e3i)T^{2} \) |
| 79 | \( 1 + (23.3 - 58.6i)T + (-4.53e3 - 4.29e3i)T^{2} \) |
| 83 | \( 1 + (9.41 + 20.3i)T + (-4.45e3 + 5.25e3i)T^{2} \) |
| 89 | \( 1 + (-22.7 + 37.8i)T + (-3.71e3 - 6.99e3i)T^{2} \) |
| 97 | \( 1 + (-0.456 - 0.537i)T + (-1.52e3 + 9.28e3i)T^{2} \) |
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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
−12.01886476202051673958597026773, −10.98327083563638217344518051964, −9.896887055667651809570156707033, −8.890476819771194349557497792209, −8.244229557405200067724388627211, −7.74854265191463467125233930126, −4.67134513709665178735653967231, −3.90911953281365363294001758290, −1.88988974992846127842429876691, −1.17415952488627133644309227955,
2.22432912843238416189359128855, 4.40978959146782494087692255994, 5.83100987513857393672154317674, 7.16072574981219929281562457368, 7.76037493805381455900703037954, 8.564372163195463781777926631542, 9.574482450150783996825576834647, 10.58840194247259849532113023432, 11.51312625607171662086710393548, 13.81663446240508744390281338181
