L(s) = 1 | + (−0.766 + 1.32i)7-s + (−0.439 + 0.524i)13-s + (−0.5 + 0.866i)19-s + (−0.766 − 0.642i)25-s + (−1.70 − 0.984i)31-s + 1.96i·37-s + (0.326 + 0.118i)43-s + (−0.673 − 1.16i)49-s + (−1.76 + 0.642i)61-s + (1.26 + 0.223i)67-s + (−1.43 + 1.20i)73-s + (0.826 + 0.984i)79-s + (−0.358 − 0.984i)91-s + (1.70 − 0.300i)97-s + (0.592 − 0.342i)103-s + ⋯ |
L(s) = 1 | + (−0.766 + 1.32i)7-s + (−0.439 + 0.524i)13-s + (−0.5 + 0.866i)19-s + (−0.766 − 0.642i)25-s + (−1.70 − 0.984i)31-s + 1.96i·37-s + (0.326 + 0.118i)43-s + (−0.673 − 1.16i)49-s + (−1.76 + 0.642i)61-s + (1.26 + 0.223i)67-s + (−1.43 + 1.20i)73-s + (0.826 + 0.984i)79-s + (−0.358 − 0.984i)91-s + (1.70 − 0.300i)97-s + (0.592 − 0.342i)103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.660 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.660 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6599775418\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6599775418\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
good | 5 | \( 1 + (0.766 + 0.642i)T^{2} \) |
| 7 | \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \) |
| 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 | \( 1 + (1.70 + 0.984i)T + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - 1.96iT - T^{2} \) |
| 41 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 43 | \( 1 + (-0.326 - 0.118i)T + (0.766 + 0.642i)T^{2} \) |
| 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 | \( 1 + (-1.26 - 0.223i)T + (0.939 + 0.342i)T^{2} \) |
| 71 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 73 | \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \) |
| 79 | \( 1 + (-0.826 - 0.984i)T + (-0.173 + 0.984i)T^{2} \) |
| 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
−9.355491046588216570992748504069, −8.578214537759129383679067714468, −7.88147117799635762846283696592, −6.91201706772517568374390783520, −6.09426641729857269900994523933, −5.65147729795717811349418866711, −4.58293949417983782347736193770, −3.64453450186779995115191786747, −2.65059519333702155460061645144, −1.84994584453421408842044431216,
0.39379786947555079377368709751, 1.92465077296939244059607431080, 3.20508390441640231911417072977, 3.83338636893518699278408941860, 4.75435580176048061653701780178, 5.67503059387493730455249575602, 6.55616335197641461448633478146, 7.38426794964506957885590175845, 7.62164374508824831599831794951, 8.959255915821378433759589764454