L(s) = 1 | + (1.58 + 1.14i)3-s + (−0.5 − 0.866i)5-s + (0.873 + 2.68i)9-s + (0.309 − 0.951i)11-s + (0.204 − 1.94i)15-s + (−0.0646 + 0.198i)23-s + (−0.499 + 0.866i)25-s + (−1.10 + 3.39i)27-s + (−1.08 + 0.786i)31-s + (1.58 − 1.14i)33-s + (−0.309 − 0.951i)37-s + (1.89 − 2.10i)45-s + (−1.61 − 1.17i)47-s + 49-s + (−0.5 − 0.363i)53-s + ⋯ |
L(s) = 1 | + (1.58 + 1.14i)3-s + (−0.5 − 0.866i)5-s + (0.873 + 2.68i)9-s + (0.309 − 0.951i)11-s + (0.204 − 1.94i)15-s + (−0.0646 + 0.198i)23-s + (−0.499 + 0.866i)25-s + (−1.10 + 3.39i)27-s + (−1.08 + 0.786i)31-s + (1.58 − 1.14i)33-s + (−0.309 − 0.951i)37-s + (1.89 − 2.10i)45-s + (−1.61 − 1.17i)47-s + 49-s + (−0.5 − 0.363i)53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.783 - 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.783 - 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.590766545\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.590766545\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 11 | \( 1 + (-0.309 + 0.951i)T \) |
good | 3 | \( 1 + (-1.58 - 1.14i)T + (0.309 + 0.951i)T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 13 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 17 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + (0.0646 - 0.198i)T + (-0.809 - 0.587i)T^{2} \) |
| 29 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (1.08 - 0.786i)T + (0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (1.61 + 1.17i)T + (0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (0.604 + 1.86i)T + (-0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + (1.47 - 1.07i)T + (0.309 - 0.951i)T^{2} \) |
| 71 | \( 1 + (-1.58 - 1.14i)T + (0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 89 | \( 1 + (-0.564 + 1.73i)T + (-0.809 - 0.587i)T^{2} \) |
| 97 | \( 1 + (1.08 + 0.786i)T + (0.309 + 0.951i)T^{2} \) |
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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
−9.878541706414430796694513460732, −9.140264297217839502726045868682, −8.627373941793174635642108307407, −8.069996868194573985700488592692, −7.17117091453817245846020143814, −5.52558717328011938729801539270, −4.73888725886720569479095735400, −3.77294447858307119124118709170, −3.28486306474004199254339630574, −1.85940262462631432300436045114,
1.65697081987161917479303215584, 2.61947674019799653979113445956, 3.45896928822767482876833772842, 4.36494773804077246614441632900, 6.22628080896827549335875127382, 6.88489128098119817305796182381, 7.59666778860703597920945463014, 8.046974264109641573019447107835, 9.098504520237145983917082854752, 9.666544596733345526210689076567