| L(s) = 1 | − 0.517i·5-s − 2.92i·7-s − 5.64·11-s + 4.46·13-s + 2.31i·17-s + 5.06i·19-s + 1.51·23-s + 4.73·25-s − 4.76i·29-s + 7.98i·31-s − 1.51·35-s − 0.267·37-s + 8.10i·41-s + 2.92i·43-s − 4.13·47-s + ⋯ |
| L(s) = 1 | − 0.231i·5-s − 1.10i·7-s − 1.70·11-s + 1.23·13-s + 0.560i·17-s + 1.16i·19-s + 0.315·23-s + 0.946·25-s − 0.883i·29-s + 1.43i·31-s − 0.255·35-s − 0.0440·37-s + 1.26i·41-s + 0.445i·43-s − 0.602·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.713331712\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.713331712\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 0.517iT - 5T^{2} \) |
| 7 | \( 1 + 2.92iT - 7T^{2} \) |
| 11 | \( 1 + 5.64T + 11T^{2} \) |
| 13 | \( 1 - 4.46T + 13T^{2} \) |
| 17 | \( 1 - 2.31iT - 17T^{2} \) |
| 19 | \( 1 - 5.06iT - 19T^{2} \) |
| 23 | \( 1 - 1.51T + 23T^{2} \) |
| 29 | \( 1 + 4.76iT - 29T^{2} \) |
| 31 | \( 1 - 7.98iT - 31T^{2} \) |
| 37 | \( 1 + 0.267T + 37T^{2} \) |
| 41 | \( 1 - 8.10iT - 41T^{2} \) |
| 43 | \( 1 - 2.92iT - 43T^{2} \) |
| 47 | \( 1 + 4.13T + 47T^{2} \) |
| 53 | \( 1 + 4.52iT - 53T^{2} \) |
| 59 | \( 1 + 4.13T + 59T^{2} \) |
| 61 | \( 1 - 2.26T + 61T^{2} \) |
| 67 | \( 1 + 2.92iT - 67T^{2} \) |
| 71 | \( 1 - 9.77T + 71T^{2} \) |
| 73 | \( 1 - 4.66T + 73T^{2} \) |
| 79 | \( 1 - 10.9iT - 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 + 13.5iT - 89T^{2} \) |
| 97 | \( 1 + 0.535T + 97T^{2} \) |
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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
−8.156008792882300851463587321525, −7.67383241047183906725797287929, −6.75560389567321960486190357928, −6.09217171359985984726576402693, −5.23650988726559740201079683089, −4.56189579245298916169170775440, −3.66171535558394259181555704757, −3.01894209429748805524953379396, −1.73872165070198527161729450710, −0.797275706765099965441842950747,
0.63180075974528775138833177757, 2.16793889702403876024247838052, 2.74288980349910153308075295326, 3.51645852982254522813711201700, 4.77524405873408137559408084186, 5.30023311536987353048553524654, 5.94919910704124880273074164410, 6.78919074757947779927394480368, 7.51929953655179703765587970350, 8.285869902452973900868183467369
