L(s) = 1 | + (0.866 − 0.5i)2-s + (0.707 − 0.707i)3-s + (0.499 − 0.866i)4-s + (0.965 − 0.258i)5-s + (0.258 − 0.965i)6-s + (−0.258 + 0.965i)7-s − 0.999i·8-s − 1.00i·9-s + (0.707 − 0.707i)10-s + (−0.866 + 0.5i)11-s + (−0.258 − 0.965i)12-s + (−0.965 + 0.258i)13-s + (0.258 + 0.965i)14-s + (0.500 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.707 + 0.707i)17-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)2-s + (0.707 − 0.707i)3-s + (0.499 − 0.866i)4-s + (0.965 − 0.258i)5-s + (0.258 − 0.965i)6-s + (−0.258 + 0.965i)7-s − 0.999i·8-s − 1.00i·9-s + (0.707 − 0.707i)10-s + (−0.866 + 0.5i)11-s + (−0.258 − 0.965i)12-s + (−0.965 + 0.258i)13-s + (0.258 + 0.965i)14-s + (0.500 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.707 + 0.707i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.203 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.203 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.226491149\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.226491149\) |
\(L(1)\) |
|
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 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (-0.965 + 0.258i)T \) |
| 7 | \( 1 + (0.258 - 0.965i)T \) |
good | 11 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 19 | \( 1 - 1.41iT - T^{2} \) |
| 23 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \) |
| 47 | \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 53 | \( 1 + (-1 + i)T - iT^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 + iT - T^{2} \) |
| 73 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \) |
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
−9.867594805825824288891365086208, −9.021468653724841537953458938950, −8.135089906687461502475916131108, −7.03209066961274846543020939479, −6.21249663155382817659624115907, −5.55324209695534177713661018878, −4.60250619260353468037252811884, −3.27830351251094360323144998186, −2.25540237897484899668273917451, −1.86699355305746202732298483017,
2.49118594905678461955539046814, 2.88822679711101374453670714597, 4.15230331947147525137931995985, 4.96920616297144602575652087765, 5.62581831579054908708260388445, 6.97482021507375169343128272149, 7.28568748777439319181927635690, 8.422629346409857539628814376706, 9.229064062767386632646571938376, 10.11287126054719030579433835080