L(s) = 1 | + (0.788 + 2.42i)2-s + (0.332 + 0.241i)3-s + (−3.65 + 2.65i)4-s + (1.05 − 3.26i)5-s + (−0.323 + 0.996i)6-s + (−0.809 + 0.587i)7-s + (−5.18 − 3.76i)8-s + (−0.874 − 2.69i)9-s + 8.75·10-s + (−0.0297 + 3.31i)11-s − 1.85·12-s + (−0.672 − 2.06i)13-s + (−2.06 − 1.49i)14-s + (1.13 − 0.827i)15-s + (2.26 − 6.97i)16-s + (−1.39 + 4.29i)17-s + ⋯ |
L(s) = 1 | + (0.557 + 1.71i)2-s + (0.191 + 0.139i)3-s + (−1.82 + 1.32i)4-s + (0.473 − 1.45i)5-s + (−0.132 + 0.406i)6-s + (−0.305 + 0.222i)7-s + (−1.83 − 1.33i)8-s + (−0.291 − 0.897i)9-s + 2.76·10-s + (−0.00897 + 0.999i)11-s − 0.534·12-s + (−0.186 − 0.573i)13-s + (−0.551 − 0.400i)14-s + (0.294 − 0.213i)15-s + (0.566 − 1.74i)16-s + (−0.338 + 1.04i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.256 - 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.256 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.715349 + 0.929972i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.715349 + 0.929972i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (0.809 - 0.587i)T \) |
| 11 | \( 1 + (0.0297 - 3.31i)T \) |
good | 2 | \( 1 + (-0.788 - 2.42i)T + (-1.61 + 1.17i)T^{2} \) |
| 3 | \( 1 + (-0.332 - 0.241i)T + (0.927 + 2.85i)T^{2} \) |
| 5 | \( 1 + (-1.05 + 3.26i)T + (-4.04 - 2.93i)T^{2} \) |
| 13 | \( 1 + (0.672 + 2.06i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (1.39 - 4.29i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-1.96 - 1.42i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 - 0.648T + 23T^{2} \) |
| 29 | \( 1 + (-1.01 + 0.736i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (2.48 + 7.64i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (4.05 - 2.94i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-2.12 - 1.54i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 1.46T + 43T^{2} \) |
| 47 | \( 1 + (-4.07 - 2.96i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-4.14 - 12.7i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (6.20 - 4.50i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-4.42 + 13.6i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 - 6.22T + 67T^{2} \) |
| 71 | \( 1 + (-1.30 + 4.02i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-4.79 + 3.48i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-3.01 - 9.28i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (2.58 - 7.96i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 - 4.76T + 89T^{2} \) |
| 97 | \( 1 + (2.69 + 8.27i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−15.12128372372740506826941623330, −13.90844224797904833140512982090, −12.80854413370565544517451810968, −12.37718534806673325061756148757, −9.663039310984358652618323009622, −8.883048206741241640266114841760, −7.84213297160630921846679608478, −6.30508398134841114817530892232, −5.33565273809205721681149831445, −4.11910068654873997734361548081,
2.39789737135663244476015465875, 3.37342464068301906088963377283, 5.27486274977739055304281449177, 6.97228279745501848203766147613, 9.014434015017783468188348873040, 10.25500258475070453291082288791, 10.94114492581210103570143251575, 11.70995569496951734474063846622, 13.25328951786051752192671371262, 13.95543365404534595710330096619