L(s) = 1 | + (0.909 + 0.415i)2-s + (−0.989 + 0.142i)3-s + (0.654 + 0.755i)4-s + (−0.281 + 0.959i)5-s + (−0.959 − 0.281i)6-s + (0.281 + 0.959i)8-s + (0.959 − 0.281i)9-s + (−0.654 + 0.755i)10-s + (−0.755 − 0.654i)12-s + (0.142 − 0.989i)15-s + (−0.142 + 0.989i)16-s + (−0.755 − 0.345i)17-s + (0.989 + 0.142i)18-s + (0.544 + 1.19i)19-s + (−0.909 + 0.415i)20-s + ⋯ |
L(s) = 1 | + (0.909 + 0.415i)2-s + (−0.989 + 0.142i)3-s + (0.654 + 0.755i)4-s + (−0.281 + 0.959i)5-s + (−0.959 − 0.281i)6-s + (0.281 + 0.959i)8-s + (0.959 − 0.281i)9-s + (−0.654 + 0.755i)10-s + (−0.755 − 0.654i)12-s + (0.142 − 0.989i)15-s + (−0.142 + 0.989i)16-s + (−0.755 − 0.345i)17-s + (0.989 + 0.142i)18-s + (0.544 + 1.19i)19-s + (−0.909 + 0.415i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.484 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.484 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.247513202\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.247513202\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.909 - 0.415i)T \) |
| 3 | \( 1 + (0.989 - 0.142i)T \) |
| 5 | \( 1 + (0.281 - 0.959i)T \) |
| 23 | \( 1 + (-0.540 - 0.841i)T \) |
good | 7 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 11 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 13 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 17 | \( 1 + (0.755 + 0.345i)T + (0.654 + 0.755i)T^{2} \) |
| 19 | \( 1 + (-0.544 - 1.19i)T + (-0.654 + 0.755i)T^{2} \) |
| 29 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 31 | \( 1 + (1.49 + 0.215i)T + (0.959 + 0.281i)T^{2} \) |
| 37 | \( 1 + (0.841 - 0.540i)T^{2} \) |
| 41 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 43 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 47 | \( 1 + 1.68iT - T^{2} \) |
| 53 | \( 1 + (-1.27 - 1.10i)T + (0.142 + 0.989i)T^{2} \) |
| 59 | \( 1 + (-0.142 + 0.989i)T^{2} \) |
| 61 | \( 1 + (-1.07 - 0.153i)T + (0.959 + 0.281i)T^{2} \) |
| 67 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 71 | \( 1 + (0.415 + 0.909i)T^{2} \) |
| 73 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 79 | \( 1 + (-1.25 - 1.45i)T + (-0.142 + 0.989i)T^{2} \) |
| 83 | \( 1 + (-1.89 + 0.557i)T + (0.841 - 0.540i)T^{2} \) |
| 89 | \( 1 + (-0.959 + 0.281i)T^{2} \) |
| 97 | \( 1 + (0.841 + 0.540i)T^{2} \) |
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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
−10.37821693063418170562304482383, −9.350845496231048219667662219688, −8.060438418677089235854183956537, −7.16300755378827638565281056851, −6.81479096737401960922214236631, −5.76799274292919664086749079595, −5.26159013511602449643826143902, −4.04446369309659700734070447688, −3.45746666284952802351029826262, −2.04723594303268245787001326585,
0.899973405565640154167674944307, 2.18385044878299010039831263024, 3.68805528111421048195883623762, 4.66405953716067669553908264398, 5.07154279105870493101291627214, 6.00678898873066107826311942604, 6.83186729652780815653467390268, 7.61025798230922582449755980732, 8.907482397150699782368827214479, 9.615575440504286635571675103061