| L(s) = 1 | + (−0.786 − 0.618i)2-s + (1.20 − 1.69i)3-s + (0.235 + 0.971i)4-s + (2.58 − 0.497i)5-s + (−1.99 + 0.585i)6-s + (−2.50 − 0.850i)7-s + (0.415 − 0.909i)8-s + (−0.430 − 1.24i)9-s + (−2.33 − 1.20i)10-s + (4.29 − 3.38i)11-s + (1.92 + 0.772i)12-s + (−2.71 + 1.74i)13-s + (1.44 + 2.21i)14-s + (2.26 − 4.96i)15-s + (−0.888 + 0.458i)16-s + (−0.255 − 0.243i)17-s + ⋯ |
| L(s) = 1 | + (−0.555 − 0.437i)2-s + (0.695 − 0.977i)3-s + (0.117 + 0.485i)4-s + (1.15 − 0.222i)5-s + (−0.813 + 0.238i)6-s + (−0.946 − 0.321i)7-s + (0.146 − 0.321i)8-s + (−0.143 − 0.414i)9-s + (−0.738 − 0.380i)10-s + (1.29 − 1.01i)11-s + (0.556 + 0.222i)12-s + (−0.752 + 0.483i)13-s + (0.385 + 0.592i)14-s + (0.585 − 1.28i)15-s + (−0.222 + 0.114i)16-s + (−0.0619 − 0.0590i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.125 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.125 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.925139 - 1.04966i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.925139 - 1.04966i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.786 + 0.618i)T \) |
| 7 | \( 1 + (2.50 + 0.850i)T \) |
| 23 | \( 1 + (1.10 - 4.66i)T \) |
| good | 3 | \( 1 + (-1.20 + 1.69i)T + (-0.981 - 2.83i)T^{2} \) |
| 5 | \( 1 + (-2.58 + 0.497i)T + (4.64 - 1.85i)T^{2} \) |
| 11 | \( 1 + (-4.29 + 3.38i)T + (2.59 - 10.6i)T^{2} \) |
| 13 | \( 1 + (2.71 - 1.74i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (0.255 + 0.243i)T + (0.808 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-2.34 + 2.23i)T + (0.904 - 18.9i)T^{2} \) |
| 29 | \( 1 + (5.85 - 1.71i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-1.99 + 0.190i)T + (30.4 - 5.86i)T^{2} \) |
| 37 | \( 1 + (2.61 + 7.54i)T + (-29.0 + 22.8i)T^{2} \) |
| 41 | \( 1 + (5.65 + 6.52i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-1.89 - 4.14i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + (0.134 - 0.232i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.606 - 12.7i)T + (-52.7 - 5.03i)T^{2} \) |
| 59 | \( 1 + (-5.40 - 2.78i)T + (34.2 + 48.0i)T^{2} \) |
| 61 | \( 1 + (-7.22 - 10.1i)T + (-19.9 + 57.6i)T^{2} \) |
| 67 | \( 1 + (0.480 - 0.192i)T + (48.4 - 46.2i)T^{2} \) |
| 71 | \( 1 + (-0.493 + 3.43i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-2.37 - 9.80i)T + (-64.8 + 33.4i)T^{2} \) |
| 79 | \( 1 + (0.270 + 5.67i)T + (-78.6 + 7.50i)T^{2} \) |
| 83 | \( 1 + (8.81 - 10.1i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-13.5 - 1.29i)T + (87.3 + 16.8i)T^{2} \) |
| 97 | \( 1 + (6.20 + 7.15i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−11.44861967776771117209683307365, −10.20224425989974650652540961418, −9.217197292100535322999122719233, −8.985046709842869760735558897300, −7.48047866000522627028252470540, −6.79207273633334984525598050537, −5.71303574678698122765060831997, −3.69250569263458862599212802034, −2.45052408154113589686712936485, −1.27393627142557220465651566229,
2.13537914284542246269073935859, 3.51514306176303582308566297413, 4.92823575712975745558036580886, 6.20689531831929505525112407195, 6.92560140025099918521674210459, 8.416560797604903900960435886654, 9.402189897211927378685236375078, 9.871239021371287750108704901193, 10.14693667602238853671480608621, 11.83232795560076694464153091459
