| L(s) = 1 | + (0.973 + 0.230i)2-s + (−1.60 − 0.658i)3-s + (0.893 + 0.448i)4-s + (2.07 − 2.78i)5-s + (−1.40 − 1.00i)6-s + (−1.72 − 1.13i)7-s + (0.766 + 0.642i)8-s + (2.13 + 2.10i)9-s + (2.65 − 2.23i)10-s + (1.29 − 0.151i)11-s + (−1.13 − 1.30i)12-s + (0.585 − 1.95i)13-s + (−1.41 − 1.50i)14-s + (−5.15 + 3.09i)15-s + (0.597 + 0.802i)16-s + (1.17 + 6.66i)17-s + ⋯ |
| L(s) = 1 | + (0.688 + 0.163i)2-s + (−0.924 − 0.379i)3-s + (0.446 + 0.224i)4-s + (0.927 − 1.24i)5-s + (−0.574 − 0.412i)6-s + (−0.653 − 0.429i)7-s + (0.270 + 0.227i)8-s + (0.711 + 0.702i)9-s + (0.841 − 0.705i)10-s + (0.390 − 0.0455i)11-s + (−0.328 − 0.377i)12-s + (0.162 − 0.542i)13-s + (−0.379 − 0.402i)14-s + (−1.33 + 0.799i)15-s + (0.149 + 0.200i)16-s + (0.285 + 1.61i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.761 + 0.648i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.761 + 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.30586 - 0.481000i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.30586 - 0.481000i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.973 - 0.230i)T \) |
| 3 | \( 1 + (1.60 + 0.658i)T \) |
| good | 5 | \( 1 + (-2.07 + 2.78i)T + (-1.43 - 4.78i)T^{2} \) |
| 7 | \( 1 + (1.72 + 1.13i)T + (2.77 + 6.42i)T^{2} \) |
| 11 | \( 1 + (-1.29 + 0.151i)T + (10.7 - 2.53i)T^{2} \) |
| 13 | \( 1 + (-0.585 + 1.95i)T + (-10.8 - 7.14i)T^{2} \) |
| 17 | \( 1 + (-1.17 - 6.66i)T + (-15.9 + 5.81i)T^{2} \) |
| 19 | \( 1 + (-0.360 + 2.04i)T + (-17.8 - 6.49i)T^{2} \) |
| 23 | \( 1 + (4.39 - 2.89i)T + (9.10 - 21.1i)T^{2} \) |
| 29 | \( 1 + (5.65 - 5.99i)T + (-1.68 - 28.9i)T^{2} \) |
| 31 | \( 1 + (-0.391 - 6.71i)T + (-30.7 + 3.59i)T^{2} \) |
| 37 | \( 1 + (5.04 + 1.83i)T + (28.3 + 23.7i)T^{2} \) |
| 41 | \( 1 + (-11.5 + 2.72i)T + (36.6 - 18.4i)T^{2} \) |
| 43 | \( 1 + (-0.530 + 1.23i)T + (-29.5 - 31.2i)T^{2} \) |
| 47 | \( 1 + (0.169 - 2.91i)T + (-46.6 - 5.45i)T^{2} \) |
| 53 | \( 1 + (-1.23 - 2.13i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-14.3 - 1.68i)T + (57.4 + 13.6i)T^{2} \) |
| 61 | \( 1 + (-3.25 + 1.63i)T + (36.4 - 48.9i)T^{2} \) |
| 67 | \( 1 + (-6.78 - 7.19i)T + (-3.89 + 66.8i)T^{2} \) |
| 71 | \( 1 + (0.218 - 0.183i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (12.7 + 10.6i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (7.68 + 1.82i)T + (70.5 + 35.4i)T^{2} \) |
| 83 | \( 1 + (9.75 + 2.31i)T + (74.1 + 37.2i)T^{2} \) |
| 89 | \( 1 + (2.63 + 2.21i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (0.0866 + 0.116i)T + (-27.8 + 92.9i)T^{2} \) |
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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
−12.85168400490410698455196471324, −12.20903442554578131546863753753, −10.83790864894766794700544062592, −9.920950162703828529406282777118, −8.594321317257703173396068289341, −7.14980684281696518292887207058, −5.95809137578988971851497533005, −5.39251878330666399184993787330, −3.98003562871479888956450020658, −1.51769608085607746992580591471,
2.49200665585174521493340495172, 3.99017598974629230153517801293, 5.61997680836089196504093706429, 6.27006645060957711773535144763, 7.15552922412149139619397065221, 9.590224246049998104552360010546, 9.943293028889327886383819588744, 11.21658297140206791503403496833, 11.78838330142324529219629955225, 12.95971710305434908117569165642
