| L(s) = 1 | + (−0.434 − 0.502i)3-s + (1.14 − 0.165i)5-s + (11.0 + 5.04i)7-s + (1.21 − 8.47i)9-s + (4.36 − 14.8i)11-s + (6.62 + 14.4i)13-s + (−0.582 − 0.504i)15-s + (−15.4 + 24.1i)17-s + (−8.54 − 13.3i)19-s + (−2.27 − 7.74i)21-s + (19.5 − 12.0i)23-s + (−22.6 + 6.66i)25-s + (−9.81 + 6.30i)27-s + (−13.2 − 8.50i)29-s + (−28.8 + 33.3i)31-s + ⋯ |
| L(s) = 1 | + (−0.144 − 0.167i)3-s + (0.229 − 0.0330i)5-s + (1.57 + 0.721i)7-s + (0.135 − 0.941i)9-s + (0.396 − 1.35i)11-s + (0.509 + 1.11i)13-s + (−0.0388 − 0.0336i)15-s + (−0.911 + 1.41i)17-s + (−0.449 − 0.700i)19-s + (−0.108 − 0.368i)21-s + (0.851 − 0.523i)23-s + (−0.907 + 0.266i)25-s + (−0.363 + 0.233i)27-s + (−0.456 − 0.293i)29-s + (−0.931 + 1.07i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.179i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.983 + 0.179i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.45256 - 0.131325i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.45256 - 0.131325i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 + (-19.5 + 12.0i)T \) |
| good | 3 | \( 1 + (0.434 + 0.502i)T + (-1.28 + 8.90i)T^{2} \) |
| 5 | \( 1 + (-1.14 + 0.165i)T + (23.9 - 7.04i)T^{2} \) |
| 7 | \( 1 + (-11.0 - 5.04i)T + (32.0 + 37.0i)T^{2} \) |
| 11 | \( 1 + (-4.36 + 14.8i)T + (-101. - 65.4i)T^{2} \) |
| 13 | \( 1 + (-6.62 - 14.4i)T + (-110. + 127. i)T^{2} \) |
| 17 | \( 1 + (15.4 - 24.1i)T + (-120. - 262. i)T^{2} \) |
| 19 | \( 1 + (8.54 + 13.3i)T + (-149. + 328. i)T^{2} \) |
| 29 | \( 1 + (13.2 + 8.50i)T + (349. + 765. i)T^{2} \) |
| 31 | \( 1 + (28.8 - 33.3i)T + (-136. - 951. i)T^{2} \) |
| 37 | \( 1 + (29.8 + 4.29i)T + (1.31e3 + 385. i)T^{2} \) |
| 41 | \( 1 + (2.13 + 14.8i)T + (-1.61e3 + 473. i)T^{2} \) |
| 43 | \( 1 + (-35.9 + 31.1i)T + (263. - 1.83e3i)T^{2} \) |
| 47 | \( 1 + 17.3T + 2.20e3T^{2} \) |
| 53 | \( 1 + (40.8 + 18.6i)T + (1.83e3 + 2.12e3i)T^{2} \) |
| 59 | \( 1 + (0.787 + 1.72i)T + (-2.27e3 + 2.63e3i)T^{2} \) |
| 61 | \( 1 + (-41.8 - 36.3i)T + (529. + 3.68e3i)T^{2} \) |
| 67 | \( 1 + (-17.0 - 58.2i)T + (-3.77e3 + 2.42e3i)T^{2} \) |
| 71 | \( 1 + (-82.0 + 24.0i)T + (4.24e3 - 2.72e3i)T^{2} \) |
| 73 | \( 1 + (-13.8 + 8.89i)T + (2.21e3 - 4.84e3i)T^{2} \) |
| 79 | \( 1 + (95.6 - 43.6i)T + (4.08e3 - 4.71e3i)T^{2} \) |
| 83 | \( 1 + (61.2 + 8.79i)T + (6.60e3 + 1.94e3i)T^{2} \) |
| 89 | \( 1 + (62.4 - 54.1i)T + (1.12e3 - 7.84e3i)T^{2} \) |
| 97 | \( 1 + (10.9 - 1.57i)T + (9.02e3 - 2.65e3i)T^{2} \) |
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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
−13.92016211524498940378720799096, −12.66151446493968308062503525267, −11.43249251954979752905522517549, −11.00361476244223679279317997326, −8.927907791046233885015664083862, −8.593457561161246778981611854734, −6.72944908622070921275818747717, −5.63368847499134743454154690676, −4.02226452018739682116370331133, −1.71071275003215920111090240781,
1.85937364907645319887018531866, 4.35493220436536422124340695792, 5.29884490243980076418980252606, 7.25619337537468000390406053860, 8.032070217933721416480117726141, 9.615468393591160028248506032338, 10.77943523594981015482074901371, 11.42107848682652005779484149235, 12.93217791399014473203275951839, 13.88583201118241630535784012035
