| L(s) = 1 | + 0.798·3-s + (0.897 + 2.76i)5-s + (−1.99 + 1.44i)7-s − 2.36·9-s + (0.720 − 2.21i)11-s + (3.16 + 2.29i)13-s + (0.716 + 2.20i)15-s + (−1.83 + 5.64i)17-s + (−1.22 + 0.892i)19-s + (−1.58 + 1.15i)21-s + (−1.43 − 1.03i)23-s + (−2.78 + 2.02i)25-s − 4.28·27-s + (1.70 + 5.24i)29-s + (−0.744 + 2.29i)31-s + ⋯ |
| L(s) = 1 | + 0.460·3-s + (0.401 + 1.23i)5-s + (−0.752 + 0.546i)7-s − 0.787·9-s + (0.217 − 0.668i)11-s + (0.876 + 0.637i)13-s + (0.185 + 0.569i)15-s + (−0.444 + 1.36i)17-s + (−0.281 + 0.204i)19-s + (−0.346 + 0.251i)21-s + (−0.298 − 0.216i)23-s + (−0.557 + 0.404i)25-s − 0.823·27-s + (0.316 + 0.974i)29-s + (−0.133 + 0.411i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.219 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.219 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.896188 + 1.12024i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.896188 + 1.12024i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 41 | \( 1 + (-2.72 + 5.79i)T \) |
| good | 3 | \( 1 - 0.798T + 3T^{2} \) |
| 5 | \( 1 + (-0.897 - 2.76i)T + (-4.04 + 2.93i)T^{2} \) |
| 7 | \( 1 + (1.99 - 1.44i)T + (2.16 - 6.65i)T^{2} \) |
| 11 | \( 1 + (-0.720 + 2.21i)T + (-8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (-3.16 - 2.29i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (1.83 - 5.64i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (1.22 - 0.892i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (1.43 + 1.03i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-1.70 - 5.24i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (0.744 - 2.29i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (0.325 + 1.00i)T + (-29.9 + 21.7i)T^{2} \) |
| 43 | \( 1 + (4.73 + 3.43i)T + (13.2 + 40.8i)T^{2} \) |
| 47 | \( 1 + (-8.60 - 6.25i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (2.04 + 6.29i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-9.73 - 7.07i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-5.79 + 4.21i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-4.17 - 12.8i)T + (-54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (-1.91 + 5.89i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + 1.84T + 73T^{2} \) |
| 79 | \( 1 - 3.57T + 79T^{2} \) |
| 83 | \( 1 + 17.2T + 83T^{2} \) |
| 89 | \( 1 + (-13.4 + 9.75i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (4.21 + 12.9i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−10.78930857574640209866001169239, −9.998749270449128536159331432588, −8.734620335436826980172623977625, −8.619407764737355410946205745961, −7.06959755441375544749648597084, −6.22451651002850919706446700316, −5.80170692406813513507556365701, −3.85525016237783206873556800186, −3.10902779817635810758020232635, −2.09065999420170122572852245290,
0.71540993628294091576548552797, 2.38885707183032828581072694733, 3.64404473879457349060054634089, 4.74193147422123530955736375365, 5.71429461170983901819595741347, 6.69942520478612930559451599746, 7.85908117881460468556887799308, 8.667383789523330844769642029130, 9.396361298072768845521965335287, 9.978110763791731386572521773416
