L(s) = 1 | + (0.178 − 0.129i)2-s + 1.96·3-s + (−0.602 + 1.85i)4-s + (0.495 − 1.52i)5-s + (0.351 − 0.255i)6-s + (0.809 + 0.587i)7-s + (0.269 + 0.829i)8-s + 0.865·9-s + (−0.109 − 0.336i)10-s + (1.10 + 3.39i)11-s + (−1.18 + 3.64i)12-s + (3.74 − 2.71i)13-s + 0.220·14-s + (0.973 − 2.99i)15-s + (−3.00 − 2.18i)16-s + (−1.09 − 3.38i)17-s + ⋯ |
L(s) = 1 | + (0.126 − 0.0917i)2-s + 1.13·3-s + (−0.301 + 0.927i)4-s + (0.221 − 0.681i)5-s + (0.143 − 0.104i)6-s + (0.305 + 0.222i)7-s + (0.0953 + 0.293i)8-s + 0.288·9-s + (−0.0345 − 0.106i)10-s + (0.332 + 1.02i)11-s + (−0.342 + 1.05i)12-s + (1.03 − 0.753i)13-s + 0.0590·14-s + (0.251 − 0.773i)15-s + (−0.750 − 0.545i)16-s + (−0.266 − 0.820i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.292i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.956 - 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.87351 + 0.279821i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.87351 + 0.279821i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-0.809 - 0.587i)T \) |
| 41 | \( 1 + (-3.69 + 5.23i)T \) |
good | 2 | \( 1 + (-0.178 + 0.129i)T + (0.618 - 1.90i)T^{2} \) |
| 3 | \( 1 - 1.96T + 3T^{2} \) |
| 5 | \( 1 + (-0.495 + 1.52i)T + (-4.04 - 2.93i)T^{2} \) |
| 11 | \( 1 + (-1.10 - 3.39i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (-3.74 + 2.71i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (1.09 + 3.38i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-0.931 - 0.676i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (6.11 - 4.44i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (0.755 - 2.32i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (2.63 + 8.11i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-3.37 + 10.3i)T + (-29.9 - 21.7i)T^{2} \) |
| 43 | \( 1 + (4.23 - 3.07i)T + (13.2 - 40.8i)T^{2} \) |
| 47 | \( 1 + (2.41 - 1.75i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (2.46 - 7.59i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (5.76 - 4.19i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (9.61 + 6.98i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (0.0768 - 0.236i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (2.30 + 7.09i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 - 6.37T + 73T^{2} \) |
| 79 | \( 1 - 4.03T + 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 + (-3.07 - 2.23i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (4.52 - 13.9i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−12.10041960983503096592545018793, −11.07939805582020428550471954083, −9.352021358291696422361618773383, −9.165672958329205305164441546785, −7.996525521892966940475171689087, −7.52851112729050110993739734828, −5.68738283761673353309576918053, −4.37685238082501599938535117011, −3.38675567819194111963072784986, −2.05964437259910700183825073716,
1.70121217271526060587438244409, 3.24809597372865738055884888433, 4.37503423991369902148286570496, 6.01260893157253196084126868286, 6.63611310457792144364271848785, 8.294007474002547624709880328757, 8.745707307429971596529138597172, 9.871599831490917824828659642268, 10.73241292729080713557341360769, 11.50690937679745143814356173685