| L(s) = 1 | + (0.587 + 0.809i)2-s + (−2.70 − 0.879i)3-s + (−0.309 + 0.951i)4-s + (1.55 + 1.60i)5-s + (−0.879 − 2.70i)6-s − i·7-s + (−0.951 + 0.309i)8-s + (4.11 + 2.99i)9-s + (−0.382 + 2.20i)10-s + (−0.0793 + 0.0576i)11-s + (1.67 − 2.30i)12-s + (−3.14 + 4.32i)13-s + (0.809 − 0.587i)14-s + (−2.80 − 5.70i)15-s + (−0.809 − 0.587i)16-s + (−6.58 + 2.14i)17-s + ⋯ |
| L(s) = 1 | + (0.415 + 0.572i)2-s + (−1.56 − 0.507i)3-s + (−0.154 + 0.475i)4-s + (0.696 + 0.717i)5-s + (−0.358 − 1.10i)6-s − 0.377i·7-s + (−0.336 + 0.109i)8-s + (1.37 + 0.997i)9-s + (−0.120 + 0.696i)10-s + (−0.0239 + 0.0173i)11-s + (0.482 − 0.664i)12-s + (−0.871 + 1.19i)13-s + (0.216 − 0.157i)14-s + (−0.723 − 1.47i)15-s + (−0.202 − 0.146i)16-s + (−1.59 + 0.519i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.662 - 0.748i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.662 - 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.320577 + 0.711611i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.320577 + 0.711611i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.587 - 0.809i)T \) |
| 5 | \( 1 + (-1.55 - 1.60i)T \) |
| 7 | \( 1 + iT \) |
| good | 3 | \( 1 + (2.70 + 0.879i)T + (2.42 + 1.76i)T^{2} \) |
| 11 | \( 1 + (0.0793 - 0.0576i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (3.14 - 4.32i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (6.58 - 2.14i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-1.45 - 4.46i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-2.31 - 3.19i)T + (-7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (1.77 - 5.45i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (1.89 + 5.84i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-3.18 + 4.37i)T + (-11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-1.85 - 1.35i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 4.94iT - 43T^{2} \) |
| 47 | \( 1 + (5.75 + 1.87i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-12.6 - 4.10i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-6.41 - 4.66i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-4.91 + 3.57i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (4.79 - 1.55i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (-2.50 + 7.69i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (3.37 + 4.64i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (4.07 - 12.5i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (8.50 - 2.76i)T + (67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-3.12 + 2.26i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-17.3 - 5.62i)T + (78.4 + 57.0i)T^{2} \) |
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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
−11.72676069045249468521643698714, −11.13823201133679239769622051947, −10.22150393381005790962709001732, −9.128013480465413861679259388655, −7.37353929062759014306238917486, −6.89723933230787087894123484141, −6.08188736915824565328633149832, −5.25924632977324775873150684450, −4.08563649353209376344660932081, −1.98390051549657311997574859732,
0.55583748702461811725850459551, 2.55540362524959551374422670904, 4.58221065159088264136743192613, 5.04062445593670683932017769437, 5.88077648490000150049073214449, 6.88250810715966466617124848976, 8.715685438043057936170283549317, 9.689866680815885047690910373486, 10.36380320528342233081814261783, 11.28957439530858687618216254622
