| L(s) = 1 | + (−1.07 + 0.919i)2-s + (0.308 − 1.97i)4-s + (−1.05 − 0.810i)5-s + (−0.165 + 0.0444i)7-s + (1.48 + 2.40i)8-s + (1.87 − 0.100i)10-s + (−0.144 + 1.09i)11-s + (−4.15 + 0.546i)13-s + (0.137 − 0.200i)14-s + (−3.80 − 1.21i)16-s + 1.67·17-s + (3.90 + 1.61i)19-s + (−1.92 + 1.83i)20-s + (−0.853 − 1.31i)22-s + (−0.443 + 1.65i)23-s + ⋯ |
| L(s) = 1 | + (−0.759 + 0.650i)2-s + (0.154 − 0.988i)4-s + (−0.472 − 0.362i)5-s + (−0.0626 + 0.0167i)7-s + (0.525 + 0.850i)8-s + (0.594 − 0.0317i)10-s + (−0.0435 + 0.330i)11-s + (−1.15 + 0.151i)13-s + (0.0366 − 0.0535i)14-s + (−0.952 − 0.304i)16-s + 0.406·17-s + (0.895 + 0.370i)19-s + (−0.430 + 0.410i)20-s + (−0.181 − 0.279i)22-s + (−0.0924 + 0.344i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.177 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.177 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.463866 + 0.554911i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.463866 + 0.554911i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.07 - 0.919i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (1.05 + 0.810i)T + (1.29 + 4.82i)T^{2} \) |
| 7 | \( 1 + (0.165 - 0.0444i)T + (6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (0.144 - 1.09i)T + (-10.6 - 2.84i)T^{2} \) |
| 13 | \( 1 + (4.15 - 0.546i)T + (12.5 - 3.36i)T^{2} \) |
| 17 | \( 1 - 1.67T + 17T^{2} \) |
| 19 | \( 1 + (-3.90 - 1.61i)T + (13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (0.443 - 1.65i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (-3.97 - 5.17i)T + (-7.50 + 28.0i)T^{2} \) |
| 31 | \( 1 + (3.09 - 1.78i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.58 - 8.64i)T + (-26.1 + 26.1i)T^{2} \) |
| 41 | \( 1 + (-2.94 - 0.790i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (-8.75 - 1.15i)T + (41.5 + 11.1i)T^{2} \) |
| 47 | \( 1 + (-0.598 - 0.345i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.43 - 10.7i)T + (-37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (1.99 - 2.59i)T + (-15.2 - 56.9i)T^{2} \) |
| 61 | \( 1 + (7.81 - 5.99i)T + (15.7 - 58.9i)T^{2} \) |
| 67 | \( 1 + (-9.46 + 1.24i)T + (64.7 - 17.3i)T^{2} \) |
| 71 | \( 1 + (-0.107 + 0.107i)T - 71iT^{2} \) |
| 73 | \( 1 + (-3.93 - 3.93i)T + 73iT^{2} \) |
| 79 | \( 1 + (-0.777 + 1.34i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (2.96 + 3.85i)T + (-21.4 + 80.1i)T^{2} \) |
| 89 | \( 1 + (10.7 + 10.7i)T + 89iT^{2} \) |
| 97 | \( 1 + (6.90 - 11.9i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.08323906409745649664905805448, −9.548259434812976735034396071547, −8.667368113947320290300883780727, −7.72686890541065259312151555815, −7.29915197897737810766363521955, −6.19199903692554609048985835877, −5.19466507138544392602808218537, −4.39827297212004730833280752004, −2.75845703429884337478356644658, −1.20881651319179704819188235109,
0.51042767071670564559034852178, 2.27488928977339434671659429798, 3.22902521687389159044856370279, 4.22614058037026619874675803867, 5.49673881781988886665556401469, 6.86225620858672150141429325814, 7.57315868733139946214778632827, 8.181185470435123540804060260774, 9.391794320019141062765172109420, 9.779376602444212014906171553447
