| L(s) = 1 | + (0.685 − 3.44i)3-s + (4.33 − 2.89i)5-s + (−4.47 − 2.98i)7-s + (−3.10 − 1.28i)9-s + (−6.01 + 1.19i)11-s + (1.62 + 1.62i)13-s + (−7.02 − 16.9i)15-s + (5.89 − 15.9i)17-s + (0.00547 − 0.00226i)19-s + (−13.3 + 13.3i)21-s + (−3.14 − 15.8i)23-s + (0.852 − 2.05i)25-s + (11.0 − 16.4i)27-s + (30.4 + 45.6i)29-s + (26.5 + 5.27i)31-s + ⋯ |
| L(s) = 1 | + (0.228 − 1.14i)3-s + (0.867 − 0.579i)5-s + (−0.638 − 0.426i)7-s + (−0.345 − 0.143i)9-s + (−0.546 + 0.108i)11-s + (0.125 + 0.125i)13-s + (−0.468 − 1.13i)15-s + (0.346 − 0.938i)17-s + (0.000288 − 0.000119i)19-s + (−0.636 + 0.636i)21-s + (−0.136 − 0.687i)23-s + (0.0341 − 0.0823i)25-s + (0.407 − 0.610i)27-s + (1.05 + 1.57i)29-s + (0.856 + 0.170i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.100 + 0.994i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.06467 - 1.17789i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.06467 - 1.17789i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + (-5.89 + 15.9i)T \) |
| good | 3 | \( 1 + (-0.685 + 3.44i)T + (-8.31 - 3.44i)T^{2} \) |
| 5 | \( 1 + (-4.33 + 2.89i)T + (9.56 - 23.0i)T^{2} \) |
| 7 | \( 1 + (4.47 + 2.98i)T + (18.7 + 45.2i)T^{2} \) |
| 11 | \( 1 + (6.01 - 1.19i)T + (111. - 46.3i)T^{2} \) |
| 13 | \( 1 + (-1.62 - 1.62i)T + 169iT^{2} \) |
| 19 | \( 1 + (-0.00547 + 0.00226i)T + (255. - 255. i)T^{2} \) |
| 23 | \( 1 + (3.14 + 15.8i)T + (-488. + 202. i)T^{2} \) |
| 29 | \( 1 + (-30.4 - 45.6i)T + (-321. + 776. i)T^{2} \) |
| 31 | \( 1 + (-26.5 - 5.27i)T + (887. + 367. i)T^{2} \) |
| 37 | \( 1 + (6.87 - 34.5i)T + (-1.26e3 - 523. i)T^{2} \) |
| 41 | \( 1 + (-11.5 - 7.71i)T + (643. + 1.55e3i)T^{2} \) |
| 43 | \( 1 + (9.18 + 3.80i)T + (1.30e3 + 1.30e3i)T^{2} \) |
| 47 | \( 1 + (-59.1 - 59.1i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (30.5 - 12.6i)T + (1.98e3 - 1.98e3i)T^{2} \) |
| 59 | \( 1 + (36.5 - 88.2i)T + (-2.46e3 - 2.46e3i)T^{2} \) |
| 61 | \( 1 + (-48.1 + 72.1i)T + (-1.42e3 - 3.43e3i)T^{2} \) |
| 67 | \( 1 + 2.44iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (17.8 - 89.7i)T + (-4.65e3 - 1.92e3i)T^{2} \) |
| 73 | \( 1 + (-104. + 69.7i)T + (2.03e3 - 4.92e3i)T^{2} \) |
| 79 | \( 1 + (-79.9 + 15.8i)T + (5.76e3 - 2.38e3i)T^{2} \) |
| 83 | \( 1 + (57.5 + 138. i)T + (-4.87e3 + 4.87e3i)T^{2} \) |
| 89 | \( 1 + (76.4 - 76.4i)T - 7.92e3iT^{2} \) |
| 97 | \( 1 + (28.1 + 42.1i)T + (-3.60e3 + 8.69e3i)T^{2} \) |
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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
−12.82233470342598345636912447986, −12.13571730941232226407658586464, −10.51499067376717222897951974977, −9.556378872172412762661805126273, −8.387371087907022273336184400612, −7.21329984432810952067978571573, −6.31166176519682395938618585616, −4.90335466928155573447110619006, −2.75772586247782068335881215270, −1.15995349171348276974063761966,
2.59289627677425071695176784108, 3.91104761223267662676983716510, 5.48833215671515685929928780513, 6.46078515120187678762106610949, 8.167910718307290712314720428864, 9.465848777014330176586571143528, 10.05287723304354446297234077969, 10.79714156456447303809051893989, 12.27097706886710450384980258223, 13.41425662266772063290167850427
