L(s) = 1 | + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (0.707 − 0.707i)8-s + (−0.707 − 1.22i)11-s + (0.500 − 0.866i)16-s + (−1 − 0.999i)22-s + (0.707 − 1.22i)23-s + (0.5 + 0.866i)25-s + 1.41i·29-s + (0.258 − 0.965i)32-s + 2i·43-s + (−1.22 − 0.707i)44-s + (0.366 − 1.36i)46-s + (0.707 + 0.707i)50-s + (−1.22 + 0.707i)53-s + ⋯ |
L(s) = 1 | + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (0.707 − 0.707i)8-s + (−0.707 − 1.22i)11-s + (0.500 − 0.866i)16-s + (−1 − 0.999i)22-s + (0.707 − 1.22i)23-s + (0.5 + 0.866i)25-s + 1.41i·29-s + (0.258 − 0.965i)32-s + 2i·43-s + (−1.22 − 0.707i)44-s + (0.366 − 1.36i)46-s + (0.707 + 0.707i)50-s + (−1.22 + 0.707i)53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.627 + 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.627 + 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.028612162\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.028612162\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.965 + 0.258i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 - 1.41iT - T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - 2iT - T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 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 | \( 1 + 1.41T + T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.428799761603366880821295683230, −8.570698638588248276316216256421, −7.67732153791640869113703597527, −6.78208831686678674393145666608, −6.03395634063371128019131976239, −5.21370321704005017961418917439, −4.51679156479561831040157366173, −3.26795231523470101728585805068, −2.79441453195413899931943386968, −1.28190411002428933982519172968,
1.88786330712882445247689969452, 2.79765304332580955171098204528, 3.89454210711016425804671586639, 4.74637655061419750970090766859, 5.40975860610207826418637053202, 6.34435916528212467240360821044, 7.21503397241020951343478397556, 7.71191801392287446066436844166, 8.645498604320676413142903174710, 9.748964978949175800518305992141