L(s) = 1 | + (0.838 − 0.544i)2-s + (0.952 − 1.44i)3-s + (0.406 − 0.913i)4-s + (2.22 + 0.222i)5-s + (0.0114 − 1.73i)6-s + (2.68 − 0.718i)7-s + (−0.156 − 0.987i)8-s + (−1.18 − 2.75i)9-s + (1.98 − 1.02i)10-s + (−0.792 + 3.72i)11-s + (−0.933 − 1.45i)12-s + (−3.07 + 4.73i)13-s + (1.85 − 2.06i)14-s + (2.44 − 3.00i)15-s + (−0.669 − 0.743i)16-s + (−1.09 + 0.172i)17-s + ⋯ |
L(s) = 1 | + (0.593 − 0.385i)2-s + (0.550 − 0.835i)3-s + (0.203 − 0.456i)4-s + (0.995 + 0.0996i)5-s + (0.00467 − 0.707i)6-s + (1.01 − 0.271i)7-s + (−0.0553 − 0.349i)8-s + (−0.394 − 0.918i)9-s + (0.628 − 0.324i)10-s + (−0.238 + 1.12i)11-s + (−0.269 − 0.421i)12-s + (−0.852 + 1.31i)13-s + (0.496 − 0.551i)14-s + (0.630 − 0.776i)15-s + (−0.167 − 0.185i)16-s + (−0.264 + 0.0418i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.348 + 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.348 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.18972 - 1.52188i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.18972 - 1.52188i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.838 + 0.544i)T \) |
| 3 | \( 1 + (-0.952 + 1.44i)T \) |
| 5 | \( 1 + (-2.22 - 0.222i)T \) |
good | 7 | \( 1 + (-2.68 + 0.718i)T + (6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (0.792 - 3.72i)T + (-10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (3.07 - 4.73i)T + (-5.28 - 11.8i)T^{2} \) |
| 17 | \( 1 + (1.09 - 0.172i)T + (16.1 - 5.25i)T^{2} \) |
| 19 | \( 1 + (1.56 - 2.15i)T + (-5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (6.58 - 0.345i)T + (22.8 - 2.40i)T^{2} \) |
| 29 | \( 1 + (-0.176 - 1.68i)T + (-28.3 + 6.02i)T^{2} \) |
| 31 | \( 1 + (-0.874 + 8.32i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (3.47 - 1.76i)T + (21.7 - 29.9i)T^{2} \) |
| 41 | \( 1 + (0.724 + 3.40i)T + (-37.4 + 16.6i)T^{2} \) |
| 43 | \( 1 + (2.99 + 11.1i)T + (-37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (-6.70 - 5.42i)T + (9.77 + 45.9i)T^{2} \) |
| 53 | \( 1 + (11.8 + 1.88i)T + (50.4 + 16.3i)T^{2} \) |
| 59 | \( 1 + (3.20 - 0.681i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (1.78 + 0.378i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (-10.6 + 8.65i)T + (13.9 - 65.5i)T^{2} \) |
| 71 | \( 1 + (-9.24 - 12.7i)T + (-21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-6.53 - 3.33i)T + (42.9 + 59.0i)T^{2} \) |
| 79 | \( 1 + (-17.3 + 1.82i)T + (77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (-1.36 - 0.525i)T + (61.6 + 55.5i)T^{2} \) |
| 89 | \( 1 + (0.305 + 0.939i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-5.42 + 6.69i)T + (-20.1 - 94.8i)T^{2} \) |
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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
−11.06519910335491075464574404411, −9.952510172335981818624816454821, −9.312181858106971566418370357180, −8.050700274928756629692990831305, −7.11814750394109847495627987174, −6.28961803228919791076011527662, −5.06222836938904756627287632606, −4.04137072625161587731810348515, −2.12506437220377193855005067760, −1.92655048880569716844898294767,
2.25162075887671583643014645571, 3.26604267956277959430879774386, 4.81856454325532053555205991094, 5.28026431208547688101045781023, 6.29431496762548997071409668909, 7.927287859686853926719426604876, 8.371881784652730484961662819127, 9.426559871035453455504061214174, 10.44702167457685220966411976925, 11.04745129643784698598132328178