| L(s) = 1 | + (0.325 + 1.37i)2-s + (−1.01 − 1.40i)3-s + (−1.78 + 0.894i)4-s + (−2.77 − 3.16i)5-s + (1.59 − 1.85i)6-s + (−1.04 − 0.137i)7-s + (−1.81 − 2.17i)8-s + (−0.930 + 2.85i)9-s + (3.45 − 4.85i)10-s + (2.72 + 5.52i)11-s + (3.07 + 1.59i)12-s + (−0.213 − 0.627i)13-s + (−0.149 − 1.47i)14-s + (−1.61 + 7.11i)15-s + (2.39 − 3.20i)16-s + (3.81 + 3.81i)17-s + ⋯ |
| L(s) = 1 | + (0.229 + 0.973i)2-s + (−0.587 − 0.809i)3-s + (−0.894 + 0.447i)4-s + (−1.24 − 1.41i)5-s + (0.652 − 0.757i)6-s + (−0.393 − 0.0518i)7-s + (−0.641 − 0.767i)8-s + (−0.310 + 0.950i)9-s + (1.09 − 1.53i)10-s + (0.821 + 1.66i)11-s + (0.887 + 0.461i)12-s + (−0.0590 − 0.174i)13-s + (−0.0400 − 0.395i)14-s + (−0.417 + 1.83i)15-s + (0.599 − 0.800i)16-s + (0.924 + 0.924i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.276 - 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.276 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.360604 + 0.478834i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.360604 + 0.478834i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.325 - 1.37i)T \) |
| 3 | \( 1 + (1.01 + 1.40i)T \) |
| good | 5 | \( 1 + (2.77 + 3.16i)T + (-0.652 + 4.95i)T^{2} \) |
| 7 | \( 1 + (1.04 + 0.137i)T + (6.76 + 1.81i)T^{2} \) |
| 11 | \( 1 + (-2.72 - 5.52i)T + (-6.69 + 8.72i)T^{2} \) |
| 13 | \( 1 + (0.213 + 0.627i)T + (-10.3 + 7.91i)T^{2} \) |
| 17 | \( 1 + (-3.81 - 3.81i)T + 17iT^{2} \) |
| 19 | \( 1 + (-0.464 + 2.33i)T + (-17.5 - 7.27i)T^{2} \) |
| 23 | \( 1 + (-0.493 - 3.75i)T + (-22.2 + 5.95i)T^{2} \) |
| 29 | \( 1 + (4.78 - 0.313i)T + (28.7 - 3.78i)T^{2} \) |
| 31 | \( 1 + (-1.86 + 1.07i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.122 - 0.615i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (-0.296 - 2.25i)T + (-39.6 + 10.6i)T^{2} \) |
| 43 | \( 1 + (-8.00 + 3.94i)T + (26.1 - 34.1i)T^{2} \) |
| 47 | \( 1 + (6.55 - 1.75i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (6.39 - 9.57i)T + (-20.2 - 48.9i)T^{2} \) |
| 59 | \( 1 + (3.20 + 3.65i)T + (-7.70 + 58.4i)T^{2} \) |
| 61 | \( 1 + (0.171 + 2.62i)T + (-60.4 + 7.96i)T^{2} \) |
| 67 | \( 1 + (-2.49 - 1.23i)T + (40.7 + 53.1i)T^{2} \) |
| 71 | \( 1 + (-3.32 - 8.02i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (3.27 - 7.91i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (0.439 + 1.64i)T + (-68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (-10.0 - 8.78i)T + (10.8 + 82.2i)T^{2} \) |
| 89 | \( 1 + (13.7 - 5.70i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + (-6.87 - 3.96i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.40348750284453426977206964854, −9.768180363971149385817449577208, −8.997943285738905458643146863823, −7.87959058486997754046593265343, −7.55181636021630984145215880041, −6.58245325571752218741619186205, −5.43394487019761949334777457506, −4.64624454801841139941098559057, −3.76808040334177349739200413803, −1.23247821215521916557661876006,
0.40340281982039490913967112903, 3.18096723049956801043442348676, 3.38152865100253430441171502962, 4.43267313154297032296935562089, 5.81766508053727258926567742672, 6.56939505457351213747004294176, 7.945244178067342321199800386135, 9.018153376804595917083398804128, 9.929318974490043350496092785467, 10.72675405040254394402572678176
