L(s) = 1 | + (0.5 − 0.866i)2-s + (0.965 + 1.67i)3-s + (−0.499 − 0.866i)4-s + (0.866 − 1.5i)5-s + 1.93·6-s + (−0.258 − 0.965i)7-s − 0.999·8-s + (−1.36 + 2.36i)9-s + (−0.866 − 1.5i)10-s + (0.258 + 0.448i)11-s + (0.965 − 1.67i)12-s + (−0.965 − 0.258i)14-s + 3.34·15-s + (−0.5 + 0.866i)16-s + (1.36 + 2.36i)18-s + (−0.258 + 0.448i)19-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)2-s + (0.965 + 1.67i)3-s + (−0.499 − 0.866i)4-s + (0.866 − 1.5i)5-s + 1.93·6-s + (−0.258 − 0.965i)7-s − 0.999·8-s + (−1.36 + 2.36i)9-s + (−0.866 − 1.5i)10-s + (0.258 + 0.448i)11-s + (0.965 − 1.67i)12-s + (−0.965 − 0.258i)14-s + 3.34·15-s + (−0.5 + 0.866i)16-s + (1.36 + 2.36i)18-s + (−0.258 + 0.448i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.750 + 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.750 + 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.809729244\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.809729244\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (0.258 + 0.965i)T \) |
| 41 | \( 1 - T \) |
good | 3 | \( 1 + (-0.965 - 1.67i)T + (-0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.258 - 0.448i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.258 - 0.448i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + 1.93T + T^{2} \) |
| 73 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.965 + 1.67i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 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
−9.964040532992200261950076304301, −9.272396752218240182718267893067, −8.854092873896675216444825801501, −7.86403190659206193508086470802, −6.04116061138301736448240791473, −5.13911413094064660381673666426, −4.42470825690778690460718283740, −3.99918958122409024678437666689, −2.82038984406950722036720502042, −1.59438215829365326960239542714,
2.10764208365349026658219098080, 2.81962520485208165693112483111, 3.50189059843053781102807040703, 5.54254930400929333986586177999, 6.31978175981018118736023005523, 6.62059304672047393159878811757, 7.40899424755949945930532328517, 8.232629488128721907264115632162, 9.006882190222763517654815996452, 9.644563265218443843544651010613