L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.5 + 0.866i)3-s + (0.499 − 0.866i)4-s − i·5-s + (−0.866 − 0.499i)6-s + 0.999i·8-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)10-s + (0.866 − 0.5i)11-s + 0.999·12-s + 13-s + (0.866 − 0.5i)15-s + (−0.5 − 0.866i)16-s − 0.999i·18-s + (−0.866 − 0.499i)20-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.5 + 0.866i)3-s + (0.499 − 0.866i)4-s − i·5-s + (−0.866 − 0.499i)6-s + 0.999i·8-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)10-s + (0.866 − 0.5i)11-s + 0.999·12-s + 13-s + (0.866 − 0.5i)15-s + (−0.5 − 0.866i)16-s − 0.999i·18-s + (−0.866 − 0.499i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 - 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 - 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9385762167\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9385762167\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 5 | \( 1 + iT \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + 1.73iT - T^{2} \) |
| 37 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 - iT - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
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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
−9.428519944071580907051698187476, −8.870082181200963869882888897449, −8.372034355589957277742947362708, −7.69928082763020614707185682121, −6.30630499702309665938167186923, −5.80709278743815082680386530166, −4.65560123037762148718833491252, −3.96570368644413284660659492057, −2.53243022209966109439006292743, −1.14443759536289464974120539366,
1.36274078236633075997521077275, 2.27762895805750125527464955583, 3.33423933166108371963427203566, 3.95628963074027665640315371396, 5.96288141344355515965707928039, 6.67085040090931358860414144844, 7.21516919212762012480718421770, 8.099167677494231285040073208022, 8.646420308462960873973441205218, 9.668223901845045325762117862641