| L(s) = 1 | + (0.686 − 0.727i)2-s + (−0.301 − 1.70i)3-s + (−0.0581 − 0.998i)4-s + (1.83 + 0.214i)5-s + (−1.44 − 0.950i)6-s + (−0.0854 + 0.0428i)7-s + (−0.766 − 0.642i)8-s + (−2.81 + 1.03i)9-s + (1.41 − 1.18i)10-s + (0.254 − 0.589i)11-s + (−1.68 + 0.400i)12-s + (−1.34 + 0.319i)13-s + (−0.0274 + 0.0915i)14-s + (−0.188 − 3.19i)15-s + (−0.993 + 0.116i)16-s + (0.845 + 4.79i)17-s + ⋯ |
| L(s) = 1 | + (0.485 − 0.514i)2-s + (−0.174 − 0.984i)3-s + (−0.0290 − 0.499i)4-s + (0.821 + 0.0960i)5-s + (−0.591 − 0.388i)6-s + (−0.0322 + 0.0162i)7-s + (−0.270 − 0.227i)8-s + (−0.939 + 0.343i)9-s + (0.448 − 0.376i)10-s + (0.0767 − 0.177i)11-s + (−0.486 + 0.115i)12-s + (−0.373 + 0.0885i)13-s + (−0.00732 + 0.0244i)14-s + (−0.0486 − 0.826i)15-s + (−0.248 + 0.0290i)16-s + (0.205 + 1.16i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0165 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0165 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.01954 - 1.03660i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01954 - 1.03660i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.686 + 0.727i)T \) |
| 3 | \( 1 + (0.301 + 1.70i)T \) |
| good | 5 | \( 1 + (-1.83 - 0.214i)T + (4.86 + 1.15i)T^{2} \) |
| 7 | \( 1 + (0.0854 - 0.0428i)T + (4.18 - 5.61i)T^{2} \) |
| 11 | \( 1 + (-0.254 + 0.589i)T + (-7.54 - 8.00i)T^{2} \) |
| 13 | \( 1 + (1.34 - 0.319i)T + (11.6 - 5.83i)T^{2} \) |
| 17 | \( 1 + (-0.845 - 4.79i)T + (-15.9 + 5.81i)T^{2} \) |
| 19 | \( 1 + (-0.680 + 3.85i)T + (-17.8 - 6.49i)T^{2} \) |
| 23 | \( 1 + (-6.15 - 3.08i)T + (13.7 + 18.4i)T^{2} \) |
| 29 | \( 1 + (-1.82 - 6.08i)T + (-24.2 + 15.9i)T^{2} \) |
| 31 | \( 1 + (-0.00237 + 0.00156i)T + (12.2 - 28.4i)T^{2} \) |
| 37 | \( 1 + (-4.08 - 1.48i)T + (28.3 + 23.7i)T^{2} \) |
| 41 | \( 1 + (-5.48 - 5.81i)T + (-2.38 + 40.9i)T^{2} \) |
| 43 | \( 1 + (6.66 + 8.94i)T + (-12.3 + 41.1i)T^{2} \) |
| 47 | \( 1 + (7.83 + 5.15i)T + (18.6 + 43.1i)T^{2} \) |
| 53 | \( 1 + (5.22 + 9.04i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.22 - 2.83i)T + (-40.4 + 42.9i)T^{2} \) |
| 61 | \( 1 + (-0.152 + 2.61i)T + (-60.5 - 7.08i)T^{2} \) |
| 67 | \( 1 + (2.24 - 7.48i)T + (-55.9 - 36.8i)T^{2} \) |
| 71 | \( 1 + (6.01 - 5.04i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (5.97 + 5.01i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-7.38 + 7.82i)T + (-4.59 - 78.8i)T^{2} \) |
| 83 | \( 1 + (1.46 - 1.54i)T + (-4.82 - 82.8i)T^{2} \) |
| 89 | \( 1 + (3.99 + 3.35i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (16.4 - 1.91i)T + (94.3 - 22.3i)T^{2} \) |
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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
−12.86941843897302290293406185070, −11.71378941014149464649669560038, −10.87799831661095450220578360950, −9.716007120414654376050608146309, −8.537194176915103782346453718978, −7.07471236372562025794388005927, −6.10318397737196861875836239841, −5.05893971616537352572284284806, −3.04043683460499184371714797251, −1.61203687055671515347359088388,
2.90096710895502153972384092494, 4.48110640114763917974796793336, 5.42489671824064235418457653745, 6.43617979090827547523258158762, 7.908143496750262631010723413374, 9.293945443727560315434749202955, 9.879874264117062905721515552043, 11.13996067568577300437728914169, 12.16880077531046549933269361191, 13.31429054763881762343011959671
