| L(s) = 1 | + (1.26 − 1.89i)3-s + (1.74 + 8.74i)5-s + (−0.705 + 3.54i)7-s + (1.45 + 3.50i)9-s + (4.28 − 2.86i)11-s + (−1.13 + 1.13i)13-s + (18.8 + 7.78i)15-s + (5.25 − 16.1i)17-s + (7.58 − 18.3i)19-s + (5.83 + 5.83i)21-s + (9.73 + 14.5i)23-s + (−50.4 + 20.8i)25-s + (28.6 + 5.69i)27-s + (−12.2 + 2.43i)29-s + (−16.8 − 11.2i)31-s + ⋯ |
| L(s) = 1 | + (0.422 − 0.632i)3-s + (0.348 + 1.74i)5-s + (−0.100 + 0.506i)7-s + (0.161 + 0.389i)9-s + (0.389 − 0.260i)11-s + (−0.0874 + 0.0874i)13-s + (1.25 + 0.519i)15-s + (0.309 − 0.951i)17-s + (0.399 − 0.964i)19-s + (0.277 + 0.277i)21-s + (0.423 + 0.633i)23-s + (−2.01 + 0.835i)25-s + (1.06 + 0.210i)27-s + (−0.421 + 0.0839i)29-s + (−0.544 − 0.363i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.804 - 0.593i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.804 - 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.60859 + 0.529244i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.60859 + 0.529244i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + (-5.25 + 16.1i)T \) |
| good | 3 | \( 1 + (-1.26 + 1.89i)T + (-3.44 - 8.31i)T^{2} \) |
| 5 | \( 1 + (-1.74 - 8.74i)T + (-23.0 + 9.56i)T^{2} \) |
| 7 | \( 1 + (0.705 - 3.54i)T + (-45.2 - 18.7i)T^{2} \) |
| 11 | \( 1 + (-4.28 + 2.86i)T + (46.3 - 111. i)T^{2} \) |
| 13 | \( 1 + (1.13 - 1.13i)T - 169iT^{2} \) |
| 19 | \( 1 + (-7.58 + 18.3i)T + (-255. - 255. i)T^{2} \) |
| 23 | \( 1 + (-9.73 - 14.5i)T + (-202. + 488. i)T^{2} \) |
| 29 | \( 1 + (12.2 - 2.43i)T + (776. - 321. i)T^{2} \) |
| 31 | \( 1 + (16.8 + 11.2i)T + (367. + 887. i)T^{2} \) |
| 37 | \( 1 + (22.8 - 34.1i)T + (-523. - 1.26e3i)T^{2} \) |
| 41 | \( 1 + (-4.71 + 23.7i)T + (-1.55e3 - 643. i)T^{2} \) |
| 43 | \( 1 + (28.2 + 68.1i)T + (-1.30e3 + 1.30e3i)T^{2} \) |
| 47 | \( 1 + (-40.4 + 40.4i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-38.2 + 92.2i)T + (-1.98e3 - 1.98e3i)T^{2} \) |
| 59 | \( 1 + (104. - 43.1i)T + (2.46e3 - 2.46e3i)T^{2} \) |
| 61 | \( 1 + (18.6 + 3.70i)T + (3.43e3 + 1.42e3i)T^{2} \) |
| 67 | \( 1 - 107. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (-39.9 + 59.8i)T + (-1.92e3 - 4.65e3i)T^{2} \) |
| 73 | \( 1 + (-12.8 - 64.5i)T + (-4.92e3 + 2.03e3i)T^{2} \) |
| 79 | \( 1 + (-25.3 + 16.9i)T + (2.38e3 - 5.76e3i)T^{2} \) |
| 83 | \( 1 + (68.6 + 28.4i)T + (4.87e3 + 4.87e3i)T^{2} \) |
| 89 | \( 1 + (-22.9 - 22.9i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + (-157. + 31.3i)T + (8.69e3 - 3.60e3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.49258726661174415551387519821, −11.95516636339925953640487881152, −11.07919603822643075186574964000, −10.04933897403297801552149464492, −8.896496617131183020398221621412, −7.33878566064065428566767275121, −6.91729463945711456259572995123, −5.48637588394911983655809520355, −3.26119113393643822015602749161, −2.21892896396416698339394426692,
1.29381581193782692560169128843, 3.77653118745141948643791672795, 4.72598136736413363748317666068, 6.06049860091986628924825245962, 7.81299224006093177702779826038, 8.930649345227448431450538175149, 9.534601704429618639588743600650, 10.55174251445429301275396412002, 12.28037159817275442027615642623, 12.67781114697355568435007336316
