L(s) = 1 | + (−0.194 − 0.374i)2-s + (−0.793 + 2.23i)3-s + (1.05 − 1.48i)4-s + (−1.44 − 0.627i)5-s + (0.991 − 0.136i)6-s + (1.10 + 0.671i)7-s + (−1.59 − 0.219i)8-s + (−2.02 − 1.64i)9-s + (0.0453 + 0.663i)10-s + (−2.90 + 0.813i)11-s + (2.48 + 3.52i)12-s + (−0.889 − 4.28i)13-s + (0.0372 − 0.544i)14-s + (2.54 − 2.72i)15-s + (−0.991 − 2.79i)16-s + (0.962 + 0.269i)17-s + ⋯ |
L(s) = 1 | + (−0.137 − 0.265i)2-s + (−0.458 + 1.28i)3-s + (0.525 − 0.744i)4-s + (−0.646 − 0.280i)5-s + (0.404 − 0.0556i)6-s + (0.417 + 0.253i)7-s + (−0.565 − 0.0776i)8-s + (−0.675 − 0.549i)9-s + (0.0143 + 0.209i)10-s + (−0.875 + 0.245i)11-s + (0.718 + 1.01i)12-s + (−0.246 − 1.18i)13-s + (0.00995 − 0.145i)14-s + (0.657 − 0.704i)15-s + (−0.247 − 0.697i)16-s + (0.233 + 0.0654i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.396 + 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.396 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.320607 - 0.487560i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.320607 - 0.487560i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 + (-0.962 - 0.269i)T \) |
| 47 | \( 1 + (6.80 + 0.847i)T \) |
good | 2 | \( 1 + (0.194 + 0.374i)T + (-1.15 + 1.63i)T^{2} \) |
| 3 | \( 1 + (0.793 - 2.23i)T + (-2.32 - 1.89i)T^{2} \) |
| 5 | \( 1 + (1.44 + 0.627i)T + (3.41 + 3.65i)T^{2} \) |
| 7 | \( 1 + (-1.10 - 0.671i)T + (3.22 + 6.21i)T^{2} \) |
| 11 | \( 1 + (2.90 - 0.813i)T + (9.39 - 5.71i)T^{2} \) |
| 13 | \( 1 + (0.889 + 4.28i)T + (-11.9 + 5.17i)T^{2} \) |
| 19 | \( 1 + (2.97 - 1.29i)T + (12.9 - 13.8i)T^{2} \) |
| 23 | \( 1 + (-2.83 + 5.47i)T + (-13.2 - 18.7i)T^{2} \) |
| 29 | \( 1 + (0.695 - 3.34i)T + (-26.5 - 11.5i)T^{2} \) |
| 31 | \( 1 + (0.611 + 1.72i)T + (-24.0 + 19.5i)T^{2} \) |
| 37 | \( 1 + (0.757 + 11.0i)T + (-36.6 + 5.03i)T^{2} \) |
| 41 | \( 1 + (-4.44 + 0.611i)T + (39.4 - 11.0i)T^{2} \) |
| 43 | \( 1 + (2.68 - 3.80i)T + (-14.3 - 40.5i)T^{2} \) |
| 53 | \( 1 + (5.34 - 0.734i)T + (51.0 - 14.2i)T^{2} \) |
| 59 | \( 1 + (-4.87 - 6.90i)T + (-19.7 + 55.5i)T^{2} \) |
| 61 | \( 1 + (-0.771 + 11.2i)T + (-60.4 - 8.30i)T^{2} \) |
| 67 | \( 1 + (-7.48 + 4.55i)T + (30.8 - 59.4i)T^{2} \) |
| 71 | \( 1 + (-6.36 + 12.2i)T + (-40.9 - 58.0i)T^{2} \) |
| 73 | \( 1 + (-4.03 + 3.28i)T + (14.8 - 71.4i)T^{2} \) |
| 79 | \( 1 + (-3.48 + 3.73i)T + (-5.39 - 78.8i)T^{2} \) |
| 83 | \( 1 + (11.3 - 3.17i)T + (70.9 - 43.1i)T^{2} \) |
| 89 | \( 1 + (-3.51 - 1.52i)T + (60.7 + 65.0i)T^{2} \) |
| 97 | \( 1 + (2.10 - 5.92i)T + (-75.2 - 61.2i)T^{2} \) |
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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
−10.20250763640966362929876048364, −9.478387803999895804759661304381, −8.400076117371798163074274568633, −7.57606133063606574421962050255, −6.22099634069845994511683130635, −5.25634025107745782162270314363, −4.81299766758440917187855738549, −3.56757716588315431231950805311, −2.29202861864613299542171047486, −0.30503251017056558156101044475,
1.65230040471915259340345175117, 2.84900594416782090386732335719, 4.10057001781027508457705395880, 5.46164463159162042207475994455, 6.66722798410981377771363698273, 7.02248204893014637445992805652, 7.86979998627562885672528275602, 8.280040923347847987185357110670, 9.598450845297233839818255773973, 11.01483138787880688114300082826