| L(s) = 1 | + (1.62 + 0.434i)2-s + (−0.485 + 2.96i)3-s + (−1.02 − 0.590i)4-s + (4.97 + 0.534i)5-s + (−2.07 + 4.59i)6-s + (2.65 + 0.710i)7-s + (−6.15 − 6.15i)8-s + (−8.52 − 2.87i)9-s + (7.83 + 3.02i)10-s + (−7.26 − 12.5i)11-s + (2.24 − 2.73i)12-s + (5.67 − 1.52i)13-s + (3.99 + 2.30i)14-s + (−3.99 + 14.4i)15-s + (−4.94 − 8.56i)16-s + (5.66 − 5.66i)17-s + ⋯ |
| L(s) = 1 | + (0.810 + 0.217i)2-s + (−0.161 + 0.986i)3-s + (−0.255 − 0.147i)4-s + (0.994 + 0.106i)5-s + (−0.345 + 0.765i)6-s + (0.378 + 0.101i)7-s + (−0.768 − 0.768i)8-s + (−0.947 − 0.319i)9-s + (0.783 + 0.302i)10-s + (−0.660 − 1.14i)11-s + (0.187 − 0.228i)12-s + (0.436 − 0.117i)13-s + (0.285 + 0.164i)14-s + (−0.266 + 0.963i)15-s + (−0.308 − 0.535i)16-s + (0.333 − 0.333i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.725 - 0.688i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.725 - 0.688i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.38534 + 0.552727i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.38534 + 0.552727i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.485 - 2.96i)T \) |
| 5 | \( 1 + (-4.97 - 0.534i)T \) |
| good | 2 | \( 1 + (-1.62 - 0.434i)T + (3.46 + 2i)T^{2} \) |
| 7 | \( 1 + (-2.65 - 0.710i)T + (42.4 + 24.5i)T^{2} \) |
| 11 | \( 1 + (7.26 + 12.5i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-5.67 + 1.52i)T + (146. - 84.5i)T^{2} \) |
| 17 | \( 1 + (-5.66 + 5.66i)T - 289iT^{2} \) |
| 19 | \( 1 - 30.5iT - 361T^{2} \) |
| 23 | \( 1 + (21.8 - 5.85i)T + (458. - 264.5i)T^{2} \) |
| 29 | \( 1 + (1.51 - 0.873i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (25.5 - 44.2i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-2.78 + 2.78i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + (-16.5 + 28.6i)T + (-840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-10.2 + 38.2i)T + (-1.60e3 - 924.5i)T^{2} \) |
| 47 | \( 1 + (-41.1 - 11.0i)T + (1.91e3 + 1.10e3i)T^{2} \) |
| 53 | \( 1 + (4.77 + 4.77i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (69.8 + 40.3i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (7.61 + 13.1i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-16.5 - 61.8i)T + (-3.88e3 + 2.24e3i)T^{2} \) |
| 71 | \( 1 - 45.4T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-92.0 - 92.0i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 + (-28.1 + 16.2i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (1.79 - 6.68i)T + (-5.96e3 - 3.44e3i)T^{2} \) |
| 89 | \( 1 + 35.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (59.6 + 15.9i)T + (8.14e3 + 4.70e3i)T^{2} \) |
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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
−15.65188042834652897392155481400, −14.23749976108129313426799927423, −13.99014763223460558028442296673, −12.46012015944215423485508852628, −10.82072188691751635590666960355, −9.837406924636683409277029404832, −8.578999032364695690768823392437, −5.94125720530035475603918712662, −5.32563575688796844405465560644, −3.53821009738612134286565448391,
2.33181249313500016123687671668, 4.83900182326748366818759123622, 6.12893197812723319198498609233, 7.81069498240411771609937689214, 9.298589682903038803033432689562, 11.06513889667453108688993440864, 12.41805902504913712684630337192, 13.16107370478369931016179117435, 13.90854918234220856087123270486, 14.99650198659681102375295847536
