| L(s) = 1 | + (0.957 + 2.49i)2-s + (−0.669 + 0.743i)3-s + (−3.81 + 3.43i)4-s + (−2.91 + 0.462i)5-s + (−2.49 − 0.957i)6-s + (1.75 + 0.0921i)7-s + (−7.45 − 3.79i)8-s + (−0.104 − 0.994i)9-s + (−3.94 − 6.83i)10-s + (3.31 − 0.0362i)11-s − 5.13i·12-s + (−3.56 − 0.565i)13-s + (1.45 + 4.47i)14-s + (1.60 − 2.47i)15-s + (1.26 − 12.0i)16-s + (−3.35 + 1.49i)17-s + ⋯ |
| L(s) = 1 | + (0.676 + 1.76i)2-s + (−0.386 + 0.429i)3-s + (−1.90 + 1.71i)4-s + (−1.30 + 0.206i)5-s + (−1.01 − 0.390i)6-s + (0.664 + 0.0348i)7-s + (−2.63 − 1.34i)8-s + (−0.0348 − 0.331i)9-s + (−1.24 − 2.16i)10-s + (0.999 − 0.0109i)11-s − 1.48i·12-s + (−0.987 − 0.156i)13-s + (0.388 + 1.19i)14-s + (0.415 − 0.639i)15-s + (0.315 − 3.00i)16-s + (−0.814 + 0.362i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00190 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00190 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.564084 - 0.563011i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.564084 - 0.563011i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.669 - 0.743i)T \) |
| 11 | \( 1 + (-3.31 + 0.0362i)T \) |
| 13 | \( 1 + (3.56 + 0.565i)T \) |
| good | 2 | \( 1 + (-0.957 - 2.49i)T + (-1.48 + 1.33i)T^{2} \) |
| 5 | \( 1 + (2.91 - 0.462i)T + (4.75 - 1.54i)T^{2} \) |
| 7 | \( 1 + (-1.75 - 0.0921i)T + (6.96 + 0.731i)T^{2} \) |
| 17 | \( 1 + (3.35 - 1.49i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (-3.01 - 4.64i)T + (-7.72 + 17.3i)T^{2} \) |
| 23 | \( 1 + (3.42 - 1.97i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.264 - 1.24i)T + (-26.4 + 11.7i)T^{2} \) |
| 31 | \( 1 + (-0.313 + 1.98i)T + (-29.4 - 9.57i)T^{2} \) |
| 37 | \( 1 + (1.23 + 0.804i)T + (15.0 + 33.8i)T^{2} \) |
| 41 | \( 1 + (4.45 - 0.233i)T + (40.7 - 4.28i)T^{2} \) |
| 43 | \( 1 + (-6.20 + 10.7i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (4.30 - 8.44i)T + (-27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (9.49 - 6.89i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-0.0190 + 0.362i)T + (-58.6 - 6.16i)T^{2} \) |
| 61 | \( 1 + (-2.09 - 4.69i)T + (-40.8 + 45.3i)T^{2} \) |
| 67 | \( 1 + (-9.04 - 2.42i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (3.63 - 9.46i)T + (-52.7 - 47.5i)T^{2} \) |
| 73 | \( 1 + (-1.02 - 2.01i)T + (-42.9 + 59.0i)T^{2} \) |
| 79 | \( 1 + (-10.2 - 14.1i)T + (-24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-0.776 - 4.90i)T + (-78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + (-1.39 + 5.19i)T + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (10.7 + 8.70i)T + (20.1 + 94.8i)T^{2} \) |
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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
−12.01610619421464948888996148787, −11.26170322720734043071893443724, −9.755452090654581348196444576657, −8.673006303346850977536772389834, −7.85128281046930265921951002285, −7.22717087201567719842088893689, −6.22001395645253956749044882743, −5.17617936996036964149538175049, −4.25847627752913426163148738609, −3.64940230421604588654916107993,
0.43789634673831605733581163909, 1.93994949971403076360335980953, 3.33873390474154580909362892573, 4.54977154917896861803957029304, 4.89330577867484211586234382309, 6.59301470863575215171674126837, 7.86156784709135291195592130722, 8.954878562677672133543153620093, 9.862304433662208452413212196375, 11.13867315549202902041752467032
