| 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 | \( 1 + (-157. - 31.3i)T + (8.69e3 + 3.60e3i)T^{2} \) |
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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
−12.67781114697355568435007336316, −12.28037159817275442027615642623, −10.55174251445429301275396412002, −9.534601704429618639588743600650, −8.930649345227448431450538175149, −7.81299224006093177702779826038, −6.06049860091986628924825245962, −4.72598136736413363748317666068, −3.77653118745141948643791672795, −1.29381581193782692560169128843,
2.21892896396416698339394426692, 3.26119113393643822015602749161, 5.48637588394911983655809520355, 6.91729463945711456259572995123, 7.33878566064065428566767275121, 8.896496617131183020398221621412, 10.04933897403297801552149464492, 11.07919603822643075186574964000, 11.95516636339925953640487881152, 13.49258726661174415551387519821
