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
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.004845910\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.004845910\) |
\(L(1)\) |
|
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 | \( 1 + 0.445T + T^{2} \) |
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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
−9.398655477452377843637266016760, −8.499779999181997228505423344538, −8.237559174974487866566740401016, −7.18261231875319152485183624217, −6.45212675009432830014024313762, −5.94944807738095852363702887845, −4.83348513755775113107090671625, −3.73134309398697750689058117393, −3.01503733227752910567772794786, −1.18332793290029512309682591489,
1.19629192183085145567109304124, 1.89315450443600793012164880268, 3.13127814975886107732512491956, 3.70260687927945605092683058193, 5.20981632614323169523108200672, 5.85566723420662763715596776717, 7.05481113609609492173231792890, 7.76886022004111333665694261330, 8.467791414512692872209259085507, 9.354803515364820119312607041346