| L(s) = 1 | + (1.19 − 0.760i)2-s + (−1.08 − 1.34i)3-s + (0.844 − 1.81i)4-s + (2.35 − 0.857i)5-s + (−2.32 − 0.780i)6-s + (−2.09 + 0.368i)7-s + (−0.370 − 2.80i)8-s + (−0.632 + 2.93i)9-s + (2.15 − 2.81i)10-s + (−1.11 + 3.05i)11-s + (−3.36 + 0.834i)12-s + (2.68 − 3.19i)13-s + (−2.21 + 2.02i)14-s + (−3.72 − 2.24i)15-s + (−2.57 − 3.06i)16-s + (−0.433 − 0.250i)17-s + ⋯ |
| L(s) = 1 | + (0.843 − 0.537i)2-s + (−0.628 − 0.778i)3-s + (0.422 − 0.906i)4-s + (1.05 − 0.383i)5-s + (−0.947 − 0.318i)6-s + (−0.790 + 0.139i)7-s + (−0.131 − 0.991i)8-s + (−0.210 + 0.977i)9-s + (0.682 − 0.890i)10-s + (−0.335 + 0.921i)11-s + (−0.970 + 0.240i)12-s + (0.744 − 0.887i)13-s + (−0.591 + 0.542i)14-s + (−0.960 − 0.579i)15-s + (−0.643 − 0.765i)16-s + (−0.105 − 0.0606i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.214 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.214 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.06156 - 1.32013i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.06156 - 1.32013i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.19 + 0.760i)T \) |
| 3 | \( 1 + (1.08 + 1.34i)T \) |
| good | 5 | \( 1 + (-2.35 + 0.857i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (2.09 - 0.368i)T + (6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (1.11 - 3.05i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-2.68 + 3.19i)T + (-2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.433 + 0.250i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.69 - 4.66i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.499 + 2.83i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (0.294 - 0.247i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-1.22 - 0.216i)T + (29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-8.56 - 4.94i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.69 - 5.59i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-10.7 - 3.92i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.561 - 3.18i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + 11.9T + 53T^{2} \) |
| 59 | \( 1 + (2.08 + 5.72i)T + (-45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-4.59 + 0.810i)T + (57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (9.39 + 7.88i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-2.70 + 4.68i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.58 - 7.93i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (9.18 + 10.9i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (8.89 + 10.5i)T + (-14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (12.9 - 7.45i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.481 + 0.175i)T + (74.3 + 62.3i)T^{2} \) |
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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
−12.47996738810418392320886145057, −11.25294252524587342318436205063, −10.21591384030448501452277411870, −9.558902573126319708308085754626, −7.80269084421143854436910287179, −6.36300918374894875055637582179, −5.87148146957979430244465957669, −4.78785226080733259046444659477, −2.86669089058720222795661112983, −1.44822141215650242466982177337,
2.90398289877498454650779121900, 4.09626588037353195540020051713, 5.53972073647293888948580046782, 6.12535085723116769392632334656, 7.02848003215906515726234624711, 8.825184222467166840901802732509, 9.679460489983026413539447301321, 10.89872031997220270062689425718, 11.50643744432608599726442475423, 12.83129070777087571700947057937
