L(s) = 1 | + (−0.866 − 1.5i)2-s + (1.72 − 0.158i)3-s + (−0.5 + 0.866i)4-s + (0.358 − 2.20i)5-s + (−1.73 − 2.44i)6-s − 1.73·8-s + (2.94 − 0.548i)9-s + (−3.62 + 1.37i)10-s + (−2.44 − 1.41i)11-s + (−0.724 + 1.57i)12-s − 4·13-s + (0.267 − 3.86i)15-s + (2.49 + 4.33i)16-s + (−2.44 − 1.41i)17-s + (−3.37 − 3.94i)18-s + ⋯ |
L(s) = 1 | + (−0.612 − 1.06i)2-s + (0.995 − 0.0917i)3-s + (−0.250 + 0.433i)4-s + (0.160 − 0.987i)5-s + (−0.707 − 0.999i)6-s − 0.612·8-s + (0.983 − 0.182i)9-s + (−1.14 + 0.434i)10-s + (−0.738 − 0.426i)11-s + (−0.209 + 0.454i)12-s − 1.10·13-s + (0.0691 − 0.997i)15-s + (0.624 + 1.08i)16-s + (−0.594 − 0.342i)17-s + (−0.795 − 0.930i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.132i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0807496 + 1.21602i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0807496 + 1.21602i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.72 + 0.158i)T \) |
| 5 | \( 1 + (-0.358 + 2.20i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.866 + 1.5i)T + (-1 + 1.73i)T^{2} \) |
| 11 | \( 1 + (2.44 + 1.41i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 + (2.44 + 1.41i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.73 + 3i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 5.65iT - 29T^{2} \) |
| 31 | \( 1 + (-8.48 - 4.89i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 3.46T + 41T^{2} \) |
| 43 | \( 1 + 4.89iT - 43T^{2} \) |
| 47 | \( 1 + (-2.44 + 1.41i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.46 + 6i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-8.48 + 4.89i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.24 + 2.44i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 2.82iT - 71T^{2} \) |
| 73 | \( 1 + (4 - 6.92i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 2.82iT - 83T^{2} \) |
| 89 | \( 1 + (-5.19 - 9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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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
−9.921361478969588394840530459001, −9.181995457967315728945853859457, −8.475900557538055687564196054693, −7.83719953460635150336304932246, −6.52203765707518635120203048478, −5.17893914075333163852081385692, −4.14165473365001011741292839598, −2.76868314733855291396217333384, −2.11104269752642505430531209852, −0.64218380217712549634649815343,
2.30334134601501289000236149856, 3.08699013562282674544484558646, 4.49394396700352717281484813031, 5.81586147809904846468667798350, 6.86659973306688153530069204976, 7.43858320149163327159211072317, 8.026373617571328585707934523222, 8.980965770468230504825052139232, 9.847782547301176192033651894903, 10.29371848960383418273706672232