L(s) = 1 | + (1.15 + 1.99i)2-s + (0.736 − 1.27i)3-s + (−1.65 + 2.86i)4-s + (0.423 + 0.733i)5-s + 3.39·6-s + (1.00 − 2.44i)7-s − 3.00·8-s + (0.414 + 0.718i)9-s + (−0.975 + 1.69i)10-s + (−0.751 + 1.30i)11-s + (2.43 + 4.21i)12-s + (6.03 − 0.824i)14-s + 1.24·15-s + (−0.156 − 0.271i)16-s + (−1.03 + 1.79i)17-s + (−0.954 + 1.65i)18-s + ⋯ |
L(s) = 1 | + (0.814 + 1.41i)2-s + (0.425 − 0.736i)3-s + (−0.826 + 1.43i)4-s + (0.189 + 0.328i)5-s + 1.38·6-s + (0.378 − 0.925i)7-s − 1.06·8-s + (0.138 + 0.239i)9-s + (−0.308 + 0.534i)10-s + (−0.226 + 0.392i)11-s + (0.702 + 1.21i)12-s + (1.61 − 0.220i)14-s + 0.322·15-s + (−0.0391 − 0.0678i)16-s + (−0.251 + 0.435i)17-s + (−0.225 + 0.389i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0722 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0722 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.144314582\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.144314582\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-1.00 + 2.44i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-1.15 - 1.99i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (-0.736 + 1.27i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.423 - 0.733i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (0.751 - 1.30i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (1.03 - 1.79i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.0237 + 0.0410i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.90 - 6.77i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 1.35T + 29T^{2} \) |
| 31 | \( 1 + (-3.93 + 6.80i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.35 - 5.80i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 - 9.26T + 43T^{2} \) |
| 47 | \( 1 + (-0.180 - 0.311i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.35 - 2.34i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.820 + 1.42i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.26 + 3.91i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.02 - 1.76i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 14.2T + 71T^{2} \) |
| 73 | \( 1 + (-3.38 + 5.85i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.82 + 10.0i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 11.5T + 83T^{2} \) |
| 89 | \( 1 + (8.75 + 15.1i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 0.426T + 97T^{2} \) |
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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
−9.954918568908008575684695995800, −8.650348588447333749651225738186, −7.88271194801002048332671278513, −7.39768214720904168617385437211, −6.78807295724991049389120264437, −5.98785032299579897441657272976, −4.86229381128768755700476110477, −4.25566679356177817609911444992, −2.98716881371618523588785780621, −1.54849603125604016334418517356,
1.17572375395171569707343687934, 2.56461010715166348115906599933, 3.11284504144060928171397621881, 4.27366877909503729179694906034, 4.89763243262833667864721987493, 5.64998071708641255876923743276, 6.91309167080678318652114949232, 8.457015015872387209281606259032, 8.972711436990570777847171479117, 9.698027504653976920529661704755