| L(s) = 1 | + (0.222 − 0.974i)2-s + (0.485 + 0.149i)3-s + (−0.900 − 0.433i)4-s + (0.365 + 0.930i)5-s + (0.254 − 0.440i)6-s + (−2.29 − 3.96i)7-s + (−0.623 + 0.781i)8-s + (−2.26 − 1.54i)9-s + (0.988 − 0.149i)10-s + (2.13 − 1.03i)11-s + (−0.372 − 0.345i)12-s + (−2.32 − 0.351i)13-s + (−4.37 + 1.35i)14-s + (0.0379 + 0.506i)15-s + (0.623 + 0.781i)16-s + (1.59 − 4.07i)17-s + ⋯ |
| L(s) = 1 | + (0.157 − 0.689i)2-s + (0.280 + 0.0865i)3-s + (−0.450 − 0.216i)4-s + (0.163 + 0.416i)5-s + (0.103 − 0.179i)6-s + (−0.865 − 1.49i)7-s + (−0.220 + 0.276i)8-s + (−0.755 − 0.514i)9-s + (0.312 − 0.0471i)10-s + (0.644 − 0.310i)11-s + (−0.107 − 0.0998i)12-s + (−0.645 − 0.0973i)13-s + (−1.16 + 0.360i)14-s + (0.00980 + 0.130i)15-s + (0.155 + 0.195i)16-s + (0.387 − 0.987i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.686 + 0.726i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.686 + 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.474111 - 1.10032i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.474111 - 1.10032i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.222 + 0.974i)T \) |
| 5 | \( 1 + (-0.365 - 0.930i)T \) |
| 43 | \( 1 + (-6.39 + 1.45i)T \) |
| good | 3 | \( 1 + (-0.485 - 0.149i)T + (2.47 + 1.68i)T^{2} \) |
| 7 | \( 1 + (2.29 + 3.96i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.13 + 1.03i)T + (6.85 - 8.60i)T^{2} \) |
| 13 | \( 1 + (2.32 + 0.351i)T + (12.4 + 3.83i)T^{2} \) |
| 17 | \( 1 + (-1.59 + 4.07i)T + (-12.4 - 11.5i)T^{2} \) |
| 19 | \( 1 + (-0.837 + 0.570i)T + (6.94 - 17.6i)T^{2} \) |
| 23 | \( 1 + (-0.109 + 1.45i)T + (-22.7 - 3.42i)T^{2} \) |
| 29 | \( 1 + (-8.24 + 2.54i)T + (23.9 - 16.3i)T^{2} \) |
| 31 | \( 1 + (2.25 + 2.09i)T + (2.31 + 30.9i)T^{2} \) |
| 37 | \( 1 + (3.06 - 5.30i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.17 - 9.50i)T + (-36.9 - 17.7i)T^{2} \) |
| 47 | \( 1 + (-0.199 - 0.0959i)T + (29.3 + 36.7i)T^{2} \) |
| 53 | \( 1 + (10.3 - 1.56i)T + (50.6 - 15.6i)T^{2} \) |
| 59 | \( 1 + (-3.11 - 3.90i)T + (-13.1 + 57.5i)T^{2} \) |
| 61 | \( 1 + (-5.32 + 4.93i)T + (4.55 - 60.8i)T^{2} \) |
| 67 | \( 1 + (-8.06 + 5.49i)T + (24.4 - 62.3i)T^{2} \) |
| 71 | \( 1 + (0.374 + 4.99i)T + (-70.2 + 10.5i)T^{2} \) |
| 73 | \( 1 + (3.62 + 0.547i)T + (69.7 + 21.5i)T^{2} \) |
| 79 | \( 1 + (-7.44 - 12.8i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-10.5 - 3.25i)T + (68.5 + 46.7i)T^{2} \) |
| 89 | \( 1 + (-1.24 - 0.382i)T + (73.5 + 50.1i)T^{2} \) |
| 97 | \( 1 + (-13.9 + 6.72i)T + (60.4 - 75.8i)T^{2} \) |
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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
−10.82011430285917218392534275804, −9.829000995346011195788795611275, −9.490930870663616842345496000691, −8.155816610645229304512008332736, −6.98710475253217028697317724678, −6.20454510830849773313092994204, −4.68754690434810935827856707442, −3.53000572005083702775001736048, −2.83857149455378393573542375942, −0.69907695523762808271444246057,
2.24376601647469655749928313581, 3.51244099533447199896367036314, 5.08247329876893658446986356332, 5.78491569608151075212758426657, 6.69210407123532174295441177935, 7.934015047911213415912742189268, 8.851368707121026039721703138066, 9.256549734541132432445452391885, 10.38926560703865027662541649295, 11.90836014162046336582397356669
