L(s) = 1 | − 3i·5-s + 1.73i·7-s + 5.19·11-s − 2·13-s − 6i·17-s − 6.92i·19-s − 4·25-s − 6i·29-s + 5.19i·31-s + 5.19·35-s + 8·37-s + 10.3i·43-s − 10.3·47-s + 4·49-s + 9i·53-s + ⋯ |
L(s) = 1 | − 1.34i·5-s + 0.654i·7-s + 1.56·11-s − 0.554·13-s − 1.45i·17-s − 1.58i·19-s − 0.800·25-s − 1.11i·29-s + 0.933i·31-s + 0.878·35-s + 1.31·37-s + 1.58i·43-s − 1.51·47-s + 0.571·49-s + 1.23i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.22112 - 0.705014i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.22112 - 0.705014i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 3iT - 5T^{2} \) |
| 7 | \( 1 - 1.73iT - 7T^{2} \) |
| 11 | \( 1 - 5.19T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 + 6.92iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 - 5.19iT - 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 10.3iT - 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 - 9iT - 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 - 3.46iT - 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 - T + 73T^{2} \) |
| 79 | \( 1 - 3.46iT - 79T^{2} \) |
| 83 | \( 1 + 5.19T + 83T^{2} \) |
| 89 | \( 1 - 6iT - 89T^{2} \) |
| 97 | \( 1 + 5T + 97T^{2} \) |
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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
−11.44573115940155207031950339703, −9.651416395645334013717277303655, −9.238278483263293495798723472733, −8.543582742779613600137910644437, −7.30014750547882064027662363153, −6.24481860918258213460716285256, −5.02174898797827268788226781339, −4.40479283008061550561707377289, −2.68817947782363692601541890126, −1.00369578083096626536781980315,
1.76990784687708791770356965656, 3.44365976333848894796798811627, 4.11412279329988354133145093741, 5.90713250724622261853421486753, 6.66182779471982331097801526214, 7.44301884967625988068297462613, 8.507657893075190312983883693305, 9.797470392729814725165265279139, 10.36377766638315210562276631789, 11.21754663919364921990930186966