L(s) = 1 | + (−1.22 + 0.703i)2-s + (−1.34 − 1.08i)3-s + (1.01 − 1.72i)4-s + (1.58 − 2.74i)5-s + (2.41 + 0.388i)6-s + 3.56i·7-s + (−0.0261 + 2.82i)8-s + (0.629 + 2.93i)9-s + (−0.0138 + 4.47i)10-s + 2.09i·11-s + (−3.24 + 1.22i)12-s + (1.68 − 0.972i)13-s + (−2.50 − 4.37i)14-s + (−5.11 + 1.97i)15-s + (−1.95 − 3.48i)16-s + (3.17 + 1.83i)17-s + ⋯ |
L(s) = 1 | + (−0.867 + 0.497i)2-s + (−0.777 − 0.628i)3-s + (0.505 − 0.862i)4-s + (0.708 − 1.22i)5-s + (0.987 + 0.158i)6-s + 1.34i·7-s + (−0.00925 + 0.999i)8-s + (0.209 + 0.977i)9-s + (−0.00436 + 1.41i)10-s + 0.631i·11-s + (−0.935 + 0.353i)12-s + (0.467 − 0.269i)13-s + (−0.670 − 1.16i)14-s + (−1.32 + 0.508i)15-s + (−0.489 − 0.872i)16-s + (0.770 + 0.444i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0279i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0279i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.863480 - 0.0120770i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.863480 - 0.0120770i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.22 - 0.703i)T \) |
| 3 | \( 1 + (1.34 + 1.08i)T \) |
| 19 | \( 1 + (-4.34 - 0.328i)T \) |
good | 5 | \( 1 + (-1.58 + 2.74i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 - 3.56iT - 7T^{2} \) |
| 11 | \( 1 - 2.09iT - 11T^{2} \) |
| 13 | \( 1 + (-1.68 + 0.972i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.17 - 1.83i)T + (8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-0.948 - 1.64i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.681 + 1.18i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 8.65iT - 31T^{2} \) |
| 37 | \( 1 + 4.67iT - 37T^{2} \) |
| 41 | \( 1 + (-3.85 - 2.22i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.07 - 7.05i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.04 - 6.99i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.56 + 2.71i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.42 - 0.821i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.78 + 3.91i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.07 - 10.5i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-8.04 + 13.9i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.66 + 13.2i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (12.5 + 7.23i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 1.96iT - 83T^{2} \) |
| 89 | \( 1 + (9.31 - 5.37i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (7.48 - 12.9i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.10782176048269560593321151142, −9.790839958377129909777818432689, −9.343057879819924499149384506907, −8.284794225154026894020186292867, −7.58651337290719448148211717671, −6.15037483366042909395701034857, −5.66878998092215778214754707134, −4.96634463101109317139378215171, −2.20902616594171058429939530064, −1.14592819160512055969806366543,
1.04911895840229017057229075306, 3.06303831386395159543914066145, 3.85041168585389433775978676919, 5.49857408839433785533153966501, 6.81568888958819829534042361302, 7.07828701054249554048226151144, 8.573165543849916405403959797590, 9.842650084869045228962024418229, 10.14640733292226554258111488963, 10.95115051750739119898694159869