L(s) = 1 | + (−0.623 − 1.07i)2-s + (−0.222 + 0.385i)3-s + (−0.277 + 0.480i)4-s + 0.554·6-s − 0.554·8-s + (0.400 + 0.694i)9-s + (−0.123 − 0.213i)12-s − 1.24·13-s + (0.623 + 1.07i)16-s + (−0.900 + 1.56i)17-s + (0.5 − 0.866i)18-s + (−0.900 − 1.56i)19-s + (0.222 + 0.385i)23-s + (0.123 − 0.213i)24-s + (−0.5 + 0.866i)25-s + (0.777 + 1.34i)26-s + ⋯ |
L(s) = 1 | + (−0.623 − 1.07i)2-s + (−0.222 + 0.385i)3-s + (−0.277 + 0.480i)4-s + 0.554·6-s − 0.554·8-s + (0.400 + 0.694i)9-s + (−0.123 − 0.213i)12-s − 1.24·13-s + (0.623 + 1.07i)16-s + (−0.900 + 1.56i)17-s + (0.5 − 0.866i)18-s + (−0.900 − 1.56i)19-s + (0.222 + 0.385i)23-s + (0.123 − 0.213i)24-s + (−0.5 + 0.866i)25-s + (0.777 + 1.34i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2403303170\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2403303170\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 41 | \( 1 + T \) |
good | 2 | \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \) |
| 3 | \( 1 + (0.222 - 0.385i)T + (-0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + 1.24T + T^{2} \) |
| 17 | \( 1 + (0.900 - 1.56i)T + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.900 + 1.56i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + 1.80T + T^{2} \) |
| 47 | \( 1 + (-0.623 - 1.07i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-0.623 - 1.07i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 - 0.445T + T^{2} \) |
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
−9.643561966556845884824569782185, −9.027426783287319436826280660516, −8.275287345957190598706870319513, −7.27167007435329756697061110066, −6.44225908984818691739280741230, −5.34409398241446095882177459537, −4.55516777346898850179205652688, −3.60603968135986095328287220566, −2.38895967264862872838212690630, −1.76580778284001710353102003885,
0.20020009897710312996831977629, 2.03449118281866938440195291613, 3.24868771091844489836162223007, 4.50092349371456262817860635579, 5.39069365818424957350769252050, 6.42305358294661860649159643141, 6.80498625389605935836623737817, 7.46481482377497099278194546863, 8.298547009856418060094801826824, 8.916800549862688359453662619855