L(s) = 1 | + (−0.766 − 0.642i)3-s + (−1.11 + 0.642i)7-s + (0.173 + 0.984i)9-s + (−0.439 − 0.524i)13-s + (−0.5 − 0.866i)19-s + (1.26 + 0.223i)21-s + (0.766 − 0.642i)25-s + (0.500 − 0.866i)27-s + (0.173 + 0.300i)31-s + 1.96i·37-s + 0.684i·39-s + (0.673 + 1.85i)43-s + (0.326 − 0.565i)49-s + (−0.173 + 0.984i)57-s + (1.76 + 0.642i)61-s + ⋯ |
L(s) = 1 | + (−0.766 − 0.642i)3-s + (−1.11 + 0.642i)7-s + (0.173 + 0.984i)9-s + (−0.439 − 0.524i)13-s + (−0.5 − 0.866i)19-s + (1.26 + 0.223i)21-s + (0.766 − 0.642i)25-s + (0.500 − 0.866i)27-s + (0.173 + 0.300i)31-s + 1.96i·37-s + 0.684i·39-s + (0.673 + 1.85i)43-s + (0.326 − 0.565i)49-s + (−0.173 + 0.984i)57-s + (1.76 + 0.642i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 - 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 - 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6830994177\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6830994177\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.766 + 0.642i)T \) |
| 19 | \( 1 + (0.5 + 0.866i)T \) |
good | 5 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 7 | \( 1 + (1.11 - 0.642i)T + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.439 + 0.524i)T + (-0.173 + 0.984i)T^{2} \) |
| 17 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 23 | \( 1 + (0.766 + 0.642i)T^{2} \) |
| 29 | \( 1 + (-0.939 + 0.342i)T^{2} \) |
| 31 | \( 1 + (-0.173 - 0.300i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - 1.96iT - T^{2} \) |
| 41 | \( 1 + (0.173 + 0.984i)T^{2} \) |
| 43 | \( 1 + (-0.673 - 1.85i)T + (-0.766 + 0.642i)T^{2} \) |
| 47 | \( 1 + (-0.939 + 0.342i)T^{2} \) |
| 53 | \( 1 + (0.766 + 0.642i)T^{2} \) |
| 59 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 61 | \( 1 + (-1.76 - 0.642i)T + (0.766 + 0.642i)T^{2} \) |
| 67 | \( 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2} \) |
| 71 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 73 | \( 1 + (-1.43 - 1.20i)T + (0.173 + 0.984i)T^{2} \) |
| 79 | \( 1 + (-1.17 - 0.984i)T + (0.173 + 0.984i)T^{2} \) |
| 83 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (0.173 - 0.984i)T^{2} \) |
| 97 | \( 1 + (1.70 + 0.300i)T + (0.939 + 0.342i)T^{2} \) |
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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
−8.598478627612701960434170480464, −8.098889103175760410495451428610, −6.95444014496066658443779164700, −6.65394123686008606648149146843, −5.89328516211627623319987020166, −5.14881381096015594916653959518, −4.37027804574865133495147764724, −2.96911855463981970165082067429, −2.46361894926457198242209851449, −0.964062913499936530221524599753,
0.55025463803956389330141263975, 2.16803266547935812452378559317, 3.58089608787631806614970873336, 3.85823047578602556743672860795, 4.90286471863575881170959784598, 5.65783757840576997471363554978, 6.47567233483573727141741175623, 6.95324029408863902138180619692, 7.78551650411680079612172814678, 9.015953940434456666755799416295