L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s + (−0.5 − 0.866i)5-s + (−0.499 + 0.866i)6-s + (−0.866 − 0.5i)7-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (0.866 + 0.499i)10-s − 0.999i·12-s + 0.999·14-s + (−0.866 − 0.499i)15-s + (−0.5 − 0.866i)16-s + 0.999i·18-s − 0.999·20-s − 0.999·21-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s + (−0.5 − 0.866i)5-s + (−0.499 + 0.866i)6-s + (−0.866 − 0.5i)7-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (0.866 + 0.499i)10-s − 0.999i·12-s + 0.999·14-s + (−0.866 − 0.499i)15-s + (−0.5 − 0.866i)16-s + 0.999i·18-s − 0.999·20-s − 0.999·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6487016109\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6487016109\) |
\(L(1)\) |
|
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 | \( 1 + (-0.866 + 0.5i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (0.866 + 0.5i)T \) |
good | 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 - 1.73iT - T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( 1 - 1.73T + T^{2} \) |
| 47 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + 1.73T + T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.09424165646989701322454410118, −10.07307010933994305056149570748, −9.062992929624223481818868257377, −8.746125517597023183973744761865, −7.54251396541219015665643390175, −7.06950688701741880893781238293, −5.87729577276460478985036166441, −4.38262407465971424913251608543, −2.94431170512408659206749257327, −1.16276878335155523143065497343,
2.42066561107716752821991293931, 3.19016378566490599203851068818, 4.17617756542804213890149274598, 6.16524913211555634815121604220, 7.30732913136329681605598723490, 7.973294502695615519765131314749, 9.090473876833196123469923964105, 9.628249075660567591606041396071, 10.52486653921490050808249570467, 11.24969260612846696148660834896