L(s) = 1 | + (0.939 + 0.342i)5-s + (0.5 − 0.866i)7-s + (0.173 − 0.984i)9-s + (0.5 + 0.866i)11-s + (−0.173 − 0.984i)17-s + (−1.87 + 0.684i)23-s + (0.766 − 0.642i)35-s + (0.939 + 0.342i)43-s + (0.5 − 0.866i)45-s + (−0.173 + 0.984i)47-s + (0.173 + 0.984i)55-s + (0.939 − 0.342i)61-s + (−0.766 − 0.642i)63-s + (−0.766 + 0.642i)73-s + 0.999·77-s + ⋯ |
L(s) = 1 | + (0.939 + 0.342i)5-s + (0.5 − 0.866i)7-s + (0.173 − 0.984i)9-s + (0.5 + 0.866i)11-s + (−0.173 − 0.984i)17-s + (−1.87 + 0.684i)23-s + (0.766 − 0.642i)35-s + (0.939 + 0.342i)43-s + (0.5 − 0.866i)45-s + (−0.173 + 0.984i)47-s + (0.173 + 0.984i)55-s + (0.939 − 0.342i)61-s + (−0.766 − 0.642i)63-s + (−0.766 + 0.642i)73-s + 0.999·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.357127133\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.357127133\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 5 | \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \) |
| 7 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 17 | \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \) |
| 23 | \( 1 + (1.87 - 0.684i)T + (0.766 - 0.642i)T^{2} \) |
| 29 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 43 | \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \) |
| 47 | \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \) |
| 53 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 59 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 61 | \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \) |
| 67 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 71 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 73 | \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \) |
| 79 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 83 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 97 | \( 1 + (0.939 - 0.342i)T^{2} \) |
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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
−9.750346099088568601246495598411, −9.169139892239437205339692894111, −7.926263207608277205391561553303, −7.18301303840235298598680425120, −6.46448341307872176006544423912, −5.68296245440161214425124447464, −4.48780028805021023472260622569, −3.80771342514797373758668450991, −2.43271337586194057182518111319, −1.34147835326360555266191305555,
1.71911695996803843455250430254, 2.33897840339074086965967395856, 3.84703604604651010125958572552, 4.87090081792470773872671750546, 5.84062979180706055778506595146, 6.07859047965599710770436575432, 7.46308633260298404487336734983, 8.536953037075978065272262520588, 8.642961104448078223241792247828, 9.848508414262175129092082403631