L(s) = 1 | + 1.73i·2-s − 1.99·4-s + i·7-s − 1.73i·8-s − 1.73·11-s − 1.73·14-s + 0.999·16-s − 2.99i·22-s + 1.73i·23-s − 1.99i·28-s − 1.73·29-s − i·37-s + i·43-s + 3.46·44-s − 2.99·46-s + ⋯ |
L(s) = 1 | + 1.73i·2-s − 1.99·4-s + i·7-s − 1.73i·8-s − 1.73·11-s − 1.73·14-s + 0.999·16-s − 2.99i·22-s + 1.73i·23-s − 1.99i·28-s − 1.73·29-s − i·37-s + i·43-s + 3.46·44-s − 2.99·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5690996660\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5690996660\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 2 | \( 1 - 1.73iT - T^{2} \) |
| 11 | \( 1 + 1.73T + T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - 1.73iT - T^{2} \) |
| 29 | \( 1 + 1.73T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + iT - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - iT - T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - iT - T^{2} \) |
| 71 | \( 1 - 1.73T + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 - T + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−9.629071067753199038410813677067, −9.221814849739519169053488456851, −8.215562944474450010808414124640, −7.75774232810408546765934207659, −7.08109394419124985682647012776, −5.90907820380374481390675623890, −5.51439957514368149140919671595, −4.89914503614370000537205674676, −3.57758529889259728472204541316, −2.26010119534454783370060862338,
0.42846022477423638282803820462, 1.93883202684769553061337964284, 2.83376678204894359963705008112, 3.74285165313105776761812360269, 4.61270815220857831736936482865, 5.35172684134976090536248094010, 6.73575625616169407233442560081, 7.77289448354688003168419202219, 8.437682347195741148827930487464, 9.476046377959525770955781843709