| L(s) = 1 | + (−0.906 − 2.78i)2-s + (−0.485 + 0.352i)4-s + (10.1 − 4.74i)5-s − 23.0·7-s + (−17.5 − 12.7i)8-s + (−22.4 − 23.9i)10-s + (10.1 + 31.3i)11-s + (6.98 − 21.5i)13-s + (20.9 + 64.3i)14-s + (−21.1 + 65.0i)16-s + (−103. − 75.1i)17-s + (−64.4 − 46.8i)19-s + (−3.24 + 5.87i)20-s + (78.2 − 56.8i)22-s + (−3.36 − 10.3i)23-s + ⋯ |
| L(s) = 1 | + (−0.320 − 0.986i)2-s + (−0.0607 + 0.0441i)4-s + (0.905 − 0.424i)5-s − 1.24·7-s + (−0.775 − 0.563i)8-s + (−0.708 − 0.756i)10-s + (0.279 + 0.860i)11-s + (0.149 − 0.458i)13-s + (0.399 + 1.22i)14-s + (−0.330 + 1.01i)16-s + (−1.47 − 1.07i)17-s + (−0.778 − 0.565i)19-s + (−0.0362 + 0.0657i)20-s + (0.758 − 0.551i)22-s + (−0.0305 − 0.0939i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.807 - 0.589i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.807 - 0.589i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.239758 + 0.734805i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.239758 + 0.734805i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-10.1 + 4.74i)T \) |
| good | 2 | \( 1 + (0.906 + 2.78i)T + (-6.47 + 4.70i)T^{2} \) |
| 7 | \( 1 + 23.0T + 343T^{2} \) |
| 11 | \( 1 + (-10.1 - 31.3i)T + (-1.07e3 + 782. i)T^{2} \) |
| 13 | \( 1 + (-6.98 + 21.5i)T + (-1.77e3 - 1.29e3i)T^{2} \) |
| 17 | \( 1 + (103. + 75.1i)T + (1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (64.4 + 46.8i)T + (2.11e3 + 6.52e3i)T^{2} \) |
| 23 | \( 1 + (3.36 + 10.3i)T + (-9.84e3 + 7.15e3i)T^{2} \) |
| 29 | \( 1 + (3.28 - 2.38i)T + (7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (44.1 + 32.0i)T + (9.20e3 + 2.83e4i)T^{2} \) |
| 37 | \( 1 + (80.2 - 247. i)T + (-4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (33.2 - 102. i)T + (-5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 - 124.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (368. - 267. i)T + (3.20e4 - 9.87e4i)T^{2} \) |
| 53 | \( 1 + (-559. + 406. i)T + (4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (130. - 401. i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (125. + 385. i)T + (-1.83e5 + 1.33e5i)T^{2} \) |
| 67 | \( 1 + (484. + 352. i)T + (9.29e4 + 2.86e5i)T^{2} \) |
| 71 | \( 1 + (-734. + 533. i)T + (1.10e5 - 3.40e5i)T^{2} \) |
| 73 | \( 1 + (218. + 673. i)T + (-3.14e5 + 2.28e5i)T^{2} \) |
| 79 | \( 1 + (-69.9 + 50.8i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (-40.4 - 29.3i)T + (1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 + (202. + 624. i)T + (-5.70e5 + 4.14e5i)T^{2} \) |
| 97 | \( 1 + (-43.2 + 31.4i)T + (2.82e5 - 8.68e5i)T^{2} \) |
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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
−11.09604121606034432731836679725, −10.12897946472130762063621382147, −9.498794413858713408202869232094, −8.839742852189231699990329881873, −6.85093151664549141642837304415, −6.22570401015485694470831117870, −4.65580528253013648452669908690, −3.00458484567936525114262311002, −1.97378719785449888258594744244, −0.31892906405101810967062914846,
2.28428512384822274534699136627, 3.69376151334891368317752470283, 5.77530854112560452745907914607, 6.36205914766509518057158829379, 7.01144430811108236060746213044, 8.575061575409599766020506688871, 9.130397163668272291355102483449, 10.31875320501713699143401397980, 11.23845151339839801041306390925, 12.59013696513558495210339016570
