| L(s) = 1 | + (1.41 − 1.52i)2-s + (1.35 − 1.07i)3-s + (−0.174 − 2.32i)4-s + (−0.916 + 1.14i)5-s + (0.273 − 3.59i)6-s + (0.192 − 2.63i)7-s + (−0.536 − 0.427i)8-s + (0.675 − 2.92i)9-s + (0.455 + 3.02i)10-s + (−0.209 − 0.0477i)11-s + (−2.74 − 2.96i)12-s + (−0.777 + 0.838i)13-s + (−3.75 − 4.02i)14-s + (−0.00326 + 2.54i)15-s + (3.19 − 0.481i)16-s + (−0.231 + 3.08i)17-s + ⋯ |
| L(s) = 1 | + (1.00 − 1.07i)2-s + (0.782 − 0.622i)3-s + (−0.0870 − 1.16i)4-s + (−0.409 + 0.513i)5-s + (0.111 − 1.46i)6-s + (0.0728 − 0.997i)7-s + (−0.189 − 0.151i)8-s + (0.225 − 0.974i)9-s + (0.144 + 0.956i)10-s + (−0.0630 − 0.0143i)11-s + (−0.791 − 0.854i)12-s + (−0.215 + 0.232i)13-s + (−1.00 − 1.07i)14-s + (−0.000842 + 0.657i)15-s + (0.799 − 0.120i)16-s + (−0.0560 + 0.747i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.448 + 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.448 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.45959 - 2.36605i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.45959 - 2.36605i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.35 + 1.07i)T \) |
| 7 | \( 1 + (-0.192 + 2.63i)T \) |
| good | 2 | \( 1 + (-1.41 + 1.52i)T + (-0.149 - 1.99i)T^{2} \) |
| 5 | \( 1 + (0.916 - 1.14i)T + (-1.11 - 4.87i)T^{2} \) |
| 11 | \( 1 + (0.209 + 0.0477i)T + (9.91 + 4.77i)T^{2} \) |
| 13 | \( 1 + (0.777 - 0.838i)T + (-0.971 - 12.9i)T^{2} \) |
| 17 | \( 1 + (0.231 - 3.08i)T + (-16.8 - 2.53i)T^{2} \) |
| 19 | \( 1 + (2.52 - 1.45i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.271 + 0.562i)T + (-14.3 - 17.9i)T^{2} \) |
| 29 | \( 1 + (1.23 - 1.81i)T + (-10.5 - 26.9i)T^{2} \) |
| 31 | \( 1 + (-1.55 + 0.898i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.0841 + 0.0573i)T + (13.5 + 34.4i)T^{2} \) |
| 41 | \( 1 + (-0.0371 - 0.0946i)T + (-30.0 + 27.8i)T^{2} \) |
| 43 | \( 1 + (-0.816 + 2.08i)T + (-31.5 - 29.2i)T^{2} \) |
| 47 | \( 1 + (-7.30 - 6.77i)T + (3.51 + 46.8i)T^{2} \) |
| 53 | \( 1 + (-7.89 - 11.5i)T + (-19.3 + 49.3i)T^{2} \) |
| 59 | \( 1 + (4.49 - 11.4i)T + (-43.2 - 40.1i)T^{2} \) |
| 61 | \( 1 + (5.45 + 0.409i)T + (60.3 + 9.09i)T^{2} \) |
| 67 | \( 1 + (5.90 + 10.2i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.12 + 6.48i)T + (-44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (2.76 - 8.96i)T + (-60.3 - 41.1i)T^{2} \) |
| 79 | \( 1 + (-5.16 + 8.93i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.623 + 0.578i)T + (6.20 - 82.7i)T^{2} \) |
| 89 | \( 1 + (-10.4 + 9.71i)T + (6.65 - 88.7i)T^{2} \) |
| 97 | \( 1 + (-0.374 + 0.216i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.85168497922075306612307912842, −10.41949007601132834769333092621, −9.135431744216822398754849166282, −7.88520427716470010455198026465, −7.23582550170946020064195237832, −6.05465113923313062509235659073, −4.41067462556236546342203919245, −3.72462992539242804102193141361, −2.75048502016231068594261522418, −1.46970601971857355418144785397,
2.54187953401817435239000716304, 3.84292159700028245668098459331, 4.81077106089362928417112657793, 5.41985994550840116428442421019, 6.67759219447590092601511669294, 7.79347753788476433121628504317, 8.468969837337638439108357451430, 9.322644503208995253039197126270, 10.39887146726050257262037762634, 11.68851968611317913876864065493
