L(s) = 1 | + (1.22 + 0.707i)2-s + (−2.52 + 1.62i)3-s + (0.999 + 1.73i)4-s + (1.93 + 1.11i)5-s + (−4.23 + 0.198i)6-s + (3.85 + 5.84i)7-s + 2.82i·8-s + (3.75 − 8.18i)9-s + (1.58 + 2.73i)10-s + (−10.1 + 5.87i)11-s + (−5.33 − 2.75i)12-s + 0.292·13-s + (0.595 + 9.88i)14-s + (−6.70 + 0.314i)15-s + (−2.00 + 3.46i)16-s + (−25.0 + 14.4i)17-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (−0.841 + 0.540i)3-s + (0.249 + 0.433i)4-s + (0.387 + 0.223i)5-s + (−0.706 + 0.0331i)6-s + (0.551 + 0.834i)7-s + 0.353i·8-s + (0.416 − 0.909i)9-s + (0.158 + 0.273i)10-s + (−0.925 + 0.534i)11-s + (−0.444 − 0.229i)12-s + 0.0224·13-s + (0.0425 + 0.705i)14-s + (−0.446 + 0.0209i)15-s + (−0.125 + 0.216i)16-s + (−1.47 + 0.850i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.639 - 0.768i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.639 - 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.669756 + 1.42875i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.669756 + 1.42875i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.22 - 0.707i)T \) |
| 3 | \( 1 + (2.52 - 1.62i)T \) |
| 5 | \( 1 + (-1.93 - 1.11i)T \) |
| 7 | \( 1 + (-3.85 - 5.84i)T \) |
good | 11 | \( 1 + (10.1 - 5.87i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 - 0.292T + 169T^{2} \) |
| 17 | \( 1 + (25.0 - 14.4i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (3.00 - 5.20i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-27.2 - 15.7i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + 27.0iT - 841T^{2} \) |
| 31 | \( 1 + (-1.74 - 3.02i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-27.5 + 47.6i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 - 23.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 44.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + (2.48 + 1.43i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-79.4 + 45.8i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-67.9 + 39.2i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (11.7 - 20.3i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-35.3 - 61.2i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 103. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-33.8 - 58.6i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (65.2 - 113. i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 - 50.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (15.5 + 8.99i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + 32.2T + 9.40e3T^{2} \) |
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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
−12.61720855125443515956956121672, −11.42637765057263711763420442542, −10.84779019211353605780861032729, −9.652704607030146639156563277108, −8.508098065764041721795435767552, −7.12346705285566092214448818901, −5.98978586253445758955714478137, −5.23191601132047767937156540902, −4.18352552035262260520284981643, −2.34114327044439511218483233138,
0.813401854822189158559423182940, 2.49089426258228317926131544982, 4.53476904972338462130843649412, 5.24388354574412563512933858212, 6.52988525212841767306929257721, 7.41248145611043946508984798937, 8.803649401727849051798344721522, 10.39916690663737641654311442794, 10.93283837594316660794749787011, 11.71288828760992108182273743682