| L(s) = 1 | + (0.642 + 1.26i)2-s + (−0.828 − 5.23i)3-s + (−1.17 + 1.61i)4-s + (4.96 + 0.622i)5-s + (6.05 − 4.40i)6-s + (4.28 − 4.28i)7-s + (−2.79 − 0.442i)8-s + (−18.1 + 5.88i)9-s + (2.40 + 6.65i)10-s + (−3.02 + 9.31i)11-s + (9.43 + 4.80i)12-s + (−5.34 + 10.4i)13-s + (8.15 + 2.64i)14-s + (−0.851 − 26.4i)15-s + (−1.23 − 3.80i)16-s + (−2.20 + 13.9i)17-s + ⋯ |
| L(s) = 1 | + (0.321 + 0.630i)2-s + (−0.276 − 1.74i)3-s + (−0.293 + 0.404i)4-s + (0.992 + 0.124i)5-s + (1.00 − 0.733i)6-s + (0.612 − 0.612i)7-s + (−0.349 − 0.0553i)8-s + (−2.01 + 0.653i)9-s + (0.240 + 0.665i)10-s + (−0.275 + 0.846i)11-s + (0.786 + 0.400i)12-s + (−0.411 + 0.806i)13-s + (0.582 + 0.189i)14-s + (−0.0567 − 1.76i)15-s + (−0.0772 − 0.237i)16-s + (−0.129 + 0.819i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.891 + 0.452i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.891 + 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.28057 - 0.306411i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.28057 - 0.306411i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.642 - 1.26i)T \) |
| 5 | \( 1 + (-4.96 - 0.622i)T \) |
| good | 3 | \( 1 + (0.828 + 5.23i)T + (-8.55 + 2.78i)T^{2} \) |
| 7 | \( 1 + (-4.28 + 4.28i)T - 49iT^{2} \) |
| 11 | \( 1 + (3.02 - 9.31i)T + (-97.8 - 71.1i)T^{2} \) |
| 13 | \( 1 + (5.34 - 10.4i)T + (-99.3 - 136. i)T^{2} \) |
| 17 | \( 1 + (2.20 - 13.9i)T + (-274. - 89.3i)T^{2} \) |
| 19 | \( 1 + (2.98 + 4.11i)T + (-111. + 343. i)T^{2} \) |
| 23 | \( 1 + (-30.7 + 15.6i)T + (310. - 427. i)T^{2} \) |
| 29 | \( 1 + (-9.09 + 12.5i)T + (-259. - 799. i)T^{2} \) |
| 31 | \( 1 + (45.8 - 33.3i)T + (296. - 913. i)T^{2} \) |
| 37 | \( 1 + (26.8 + 13.6i)T + (804. + 1.10e3i)T^{2} \) |
| 41 | \( 1 + (6.72 + 20.6i)T + (-1.35e3 + 988. i)T^{2} \) |
| 43 | \( 1 + (49.7 + 49.7i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-30.8 + 4.89i)T + (2.10e3 - 682. i)T^{2} \) |
| 53 | \( 1 + (4.92 + 31.0i)T + (-2.67e3 + 868. i)T^{2} \) |
| 59 | \( 1 + (-26.1 + 8.50i)T + (2.81e3 - 2.04e3i)T^{2} \) |
| 61 | \( 1 + (6.54 - 20.1i)T + (-3.01e3 - 2.18e3i)T^{2} \) |
| 67 | \( 1 + (-8.37 + 52.8i)T + (-4.26e3 - 1.38e3i)T^{2} \) |
| 71 | \( 1 + (27.9 + 20.3i)T + (1.55e3 + 4.79e3i)T^{2} \) |
| 73 | \( 1 + (-22.0 + 11.2i)T + (3.13e3 - 4.31e3i)T^{2} \) |
| 79 | \( 1 + (19.0 - 26.1i)T + (-1.92e3 - 5.93e3i)T^{2} \) |
| 83 | \( 1 + (61.4 + 9.73i)T + (6.55e3 + 2.12e3i)T^{2} \) |
| 89 | \( 1 + (-164. - 53.5i)T + (6.40e3 + 4.65e3i)T^{2} \) |
| 97 | \( 1 + (41.3 - 6.54i)T + (8.94e3 - 2.90e3i)T^{2} \) |
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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
−14.77308693104985049848751694277, −13.97359829435940338817636114143, −13.08433787338680908923810537624, −12.27173930169938813317406213480, −10.72619680930740300330302829657, −8.761253107342656688608805225030, −7.27778239884549256389753236760, −6.66803452384914692116945759356, −5.15963705459621483409537271185, −1.91921807707773111312454351261,
3.04062903147144427441906429820, 4.98962304806499566657374162545, 5.60820003218834858429756486487, 8.773550707929941788316088483712, 9.674816473149299947605733855445, 10.70074319633138826578509271462, 11.53282009662837355695147346127, 13.17131661536271220378732603115, 14.47459743598233086749640496934, 15.22865886521976928569712050541
