| L(s) = 1 | + (−0.954 − 1.85i)3-s + (0.364 − 0.0347i)5-s + (2.37 + 1.17i)7-s + (−0.776 + 1.09i)9-s + (0.643 − 0.222i)11-s + (0.0635 + 0.216i)13-s + (−0.411 − 0.640i)15-s + (6.07 + 2.43i)17-s + (6.16 − 2.46i)19-s + (−0.0847 − 5.51i)21-s + (−0.862 − 4.71i)23-s + (−4.77 + 0.920i)25-s + (−3.42 − 0.492i)27-s + (0.0260 + 0.181i)29-s + (−4.02 + 0.191i)31-s + ⋯ |
| L(s) = 1 | + (−0.551 − 1.06i)3-s + (0.162 − 0.0155i)5-s + (0.895 + 0.444i)7-s + (−0.258 + 0.363i)9-s + (0.194 − 0.0671i)11-s + (0.0176 + 0.0599i)13-s + (−0.106 − 0.165i)15-s + (1.47 + 0.589i)17-s + (1.41 − 0.566i)19-s + (−0.0184 − 1.20i)21-s + (−0.179 − 0.983i)23-s + (−0.955 + 0.184i)25-s + (−0.659 − 0.0947i)27-s + (0.00484 + 0.0336i)29-s + (−0.722 + 0.0344i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.385 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.385 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.22214 - 0.814010i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.22214 - 0.814010i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-2.37 - 1.17i)T \) |
| 23 | \( 1 + (0.862 + 4.71i)T \) |
| good | 3 | \( 1 + (0.954 + 1.85i)T + (-1.74 + 2.44i)T^{2} \) |
| 5 | \( 1 + (-0.364 + 0.0347i)T + (4.90 - 0.946i)T^{2} \) |
| 11 | \( 1 + (-0.643 + 0.222i)T + (8.64 - 6.79i)T^{2} \) |
| 13 | \( 1 + (-0.0635 - 0.216i)T + (-10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (-6.07 - 2.43i)T + (12.3 + 11.7i)T^{2} \) |
| 19 | \( 1 + (-6.16 + 2.46i)T + (13.7 - 13.1i)T^{2} \) |
| 29 | \( 1 + (-0.0260 - 0.181i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (4.02 - 0.191i)T + (30.8 - 2.94i)T^{2} \) |
| 37 | \( 1 + (-2.69 - 1.92i)T + (12.1 + 34.9i)T^{2} \) |
| 41 | \( 1 + (7.96 + 3.63i)T + (26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-6.59 + 10.2i)T + (-17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + (-5.56 + 3.21i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (8.44 + 8.85i)T + (-2.52 + 52.9i)T^{2} \) |
| 59 | \( 1 + (-4.38 - 1.06i)T + (52.4 + 27.0i)T^{2} \) |
| 61 | \( 1 + (-11.9 - 6.13i)T + (35.3 + 49.6i)T^{2} \) |
| 67 | \( 1 + (-0.0573 - 0.297i)T + (-62.2 + 24.9i)T^{2} \) |
| 71 | \( 1 + (5.29 + 6.11i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (6.68 - 8.49i)T + (-17.2 - 70.9i)T^{2} \) |
| 79 | \( 1 + (-3.03 + 3.18i)T + (-3.75 - 78.9i)T^{2} \) |
| 83 | \( 1 + (-3.71 - 8.14i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (0.456 - 9.58i)T + (-88.5 - 8.45i)T^{2} \) |
| 97 | \( 1 + (4.29 - 9.40i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−10.51671146530636496286520278361, −9.539225009785553561012200692259, −8.466717008414389137651203338334, −7.65036763019468262229231215790, −6.93191601472394685403870481864, −5.77568704198210994409774651559, −5.28730115540591373034711623032, −3.73987366415585143926599355425, −2.14325048022881419082824203761, −1.04389513391146545197685529561,
1.40877954471713546527445704485, 3.33290309483278029708385565270, 4.29398749284023528312316355264, 5.27677847349287526705926612146, 5.80494905243213195717150053292, 7.46698284071905047994109828487, 7.891233700126881914290428260264, 9.462673475125042801822457025874, 9.802455902195353454470703702052, 10.68270110199288649751420054203
