| L(s) = 1 | + (0.142 + 0.989i)2-s + (−0.909 − 0.415i)3-s + (−0.959 + 0.281i)4-s + (−2.30 + 2.66i)5-s + (0.281 − 0.959i)6-s + (1.33 + 2.28i)7-s + (−0.415 − 0.909i)8-s + (0.654 + 0.755i)9-s + (−2.96 − 1.90i)10-s + (3.55 + 0.510i)11-s + (0.989 + 0.142i)12-s + (−3.07 + 4.78i)13-s + (−2.07 + 1.64i)14-s + (3.20 − 1.46i)15-s + (0.841 − 0.540i)16-s + (6.14 + 1.80i)17-s + ⋯ |
| L(s) = 1 | + (0.100 + 0.699i)2-s + (−0.525 − 0.239i)3-s + (−0.479 + 0.140i)4-s + (−1.03 + 1.19i)5-s + (0.115 − 0.391i)6-s + (0.503 + 0.863i)7-s + (−0.146 − 0.321i)8-s + (0.218 + 0.251i)9-s + (−0.937 − 0.602i)10-s + (1.07 + 0.154i)11-s + (0.285 + 0.0410i)12-s + (−0.853 + 1.32i)13-s + (−0.553 + 0.439i)14-s + (0.827 − 0.377i)15-s + (0.210 − 0.135i)16-s + (1.49 + 0.437i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.905 + 0.424i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.905 + 0.424i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.170851 - 0.765896i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.170851 - 0.765896i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.142 - 0.989i)T \) |
| 3 | \( 1 + (0.909 + 0.415i)T \) |
| 7 | \( 1 + (-1.33 - 2.28i)T \) |
| 23 | \( 1 + (3.28 + 3.49i)T \) |
| good | 5 | \( 1 + (2.30 - 2.66i)T + (-0.711 - 4.94i)T^{2} \) |
| 11 | \( 1 + (-3.55 - 0.510i)T + (10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (3.07 - 4.78i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-6.14 - 1.80i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (0.796 - 0.233i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (4.72 + 1.38i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (2.72 - 1.24i)T + (20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (6.97 - 6.04i)T + (5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (5.97 + 5.17i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-6.37 - 2.91i)T + (28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + 7.07iT - 47T^{2} \) |
| 53 | \( 1 + (-2.34 - 3.64i)T + (-22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-1.74 + 2.71i)T + (-24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-0.467 - 1.02i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (-5.21 + 0.749i)T + (64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (1.25 + 8.75i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (2.33 + 7.94i)T + (-61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (5.15 - 8.02i)T + (-32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-6.24 - 7.20i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (5.51 - 12.0i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (-3.40 + 3.92i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−10.54974637011801956489740912370, −9.607803249585789768663969180918, −8.610955649021964909611650007159, −7.74685857088711125426371827598, −7.02560401973254356738258949865, −6.44193770864272679513501062678, −5.45933000796254474651906242626, −4.35523753117964898081478771274, −3.50519054242232003720988065144, −1.95846243365457874742061868480,
0.41730232529448176148666701271, 1.36988556698121350273586068602, 3.48598250816467499837368537205, 4.05251530714117322199079955727, 5.03462037146984381631586122119, 5.61313927364472283441807343715, 7.34370606556487936377061416402, 7.83053122331711527381957066445, 8.821263036680703416111513282500, 9.722790478787902588661139380713
