| L(s) = 1 | + (0.443 − 1.34i)2-s + (−0.755 + 0.654i)3-s + (−1.60 − 1.19i)4-s + (3.04 − 0.437i)5-s + (0.544 + 1.30i)6-s + (−0.849 + 1.86i)7-s + (−2.31 + 1.63i)8-s + (0.142 − 0.989i)9-s + (0.761 − 4.28i)10-s + (0.297 − 1.01i)11-s + (1.99 − 0.152i)12-s + (5.65 − 2.58i)13-s + (2.12 + 1.96i)14-s + (−2.01 + 2.32i)15-s + (1.16 + 3.82i)16-s + (4.33 + 2.78i)17-s + ⋯ |
| L(s) = 1 | + (0.313 − 0.949i)2-s + (−0.436 + 0.378i)3-s + (−0.803 − 0.595i)4-s + (1.36 − 0.195i)5-s + (0.222 + 0.532i)6-s + (−0.321 + 0.703i)7-s + (−0.817 + 0.576i)8-s + (0.0474 − 0.329i)9-s + (0.240 − 1.35i)10-s + (0.0896 − 0.305i)11-s + (0.575 − 0.0441i)12-s + (1.56 − 0.715i)13-s + (0.567 + 0.525i)14-s + (−0.520 + 0.600i)15-s + (0.291 + 0.956i)16-s + (1.05 + 0.675i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.339 + 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.339 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.40663 - 0.987916i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.40663 - 0.987916i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.443 + 1.34i)T \) |
| 3 | \( 1 + (0.755 - 0.654i)T \) |
| 23 | \( 1 + (-3.49 + 3.28i)T \) |
| good | 5 | \( 1 + (-3.04 + 0.437i)T + (4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (0.849 - 1.86i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (-0.297 + 1.01i)T + (-9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (-5.65 + 2.58i)T + (8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (-4.33 - 2.78i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (3.28 + 5.11i)T + (-7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (3.04 - 4.73i)T + (-12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-1.17 + 1.35i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (-2.54 - 0.365i)T + (35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (0.885 + 6.16i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-3.44 + 2.98i)T + (6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + 1.40T + 47T^{2} \) |
| 53 | \( 1 + (12.3 + 5.65i)T + (34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (-1.90 + 0.869i)T + (38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (2.75 + 2.39i)T + (8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (-3.51 - 11.9i)T + (-56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (6.40 - 1.88i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (6.61 - 4.25i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (-5.18 - 11.3i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (4.16 + 0.599i)T + (79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-8.33 - 9.61i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (-0.988 - 6.87i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−10.71531280298205035633180182815, −9.918903263952692259939476306384, −9.047837590155743613476925468298, −8.543400288284763033229955814707, −6.38272777243743997461547920012, −5.79082033528087828378658558887, −5.12802404265393031556568190744, −3.70925936119955088042686625440, −2.60657809304384448384537893373, −1.18960832343598606056562549789,
1.45611698692849402329635022590, 3.36117408859619828921876009913, 4.57415440884058608527778406197, 5.94325959413936861340096243344, 6.11518901013332779499065020227, 7.12962679395227457678687768385, 8.042604125543513720798287831861, 9.266142034169952491476064182887, 9.870929413532618248012187004237, 10.87476339822128441091817758412
