L(s) = 1 | + (0.866 − 0.5i)3-s + (−0.515 + 1.92i)7-s + (0.499 − 0.866i)9-s + (1.28 − 0.597i)13-s + (−0.223 − 0.266i)19-s + (0.515 + 1.92i)21-s + (−0.984 + 0.173i)25-s − 0.999i·27-s + (1.48 + 1.03i)31-s + (−1.32 − 1.11i)37-s + (0.811 − 1.15i)39-s + (−1.58 − 0.424i)43-s + (−2.57 − 1.48i)49-s + (−0.326 − 0.118i)57-s + (−0.233 − 0.642i)61-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)3-s + (−0.515 + 1.92i)7-s + (0.499 − 0.866i)9-s + (1.28 − 0.597i)13-s + (−0.223 − 0.266i)19-s + (0.515 + 1.92i)21-s + (−0.984 + 0.173i)25-s − 0.999i·27-s + (1.48 + 1.03i)31-s + (−1.32 − 1.11i)37-s + (0.811 − 1.15i)39-s + (−1.58 − 0.424i)43-s + (−2.57 − 1.48i)49-s + (−0.326 − 0.118i)57-s + (−0.233 − 0.642i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 876 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0319i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 876 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0319i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.284335824\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.284335824\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 73 | \( 1 - iT \) |
good | 5 | \( 1 + (0.984 - 0.173i)T^{2} \) |
| 7 | \( 1 + (0.515 - 1.92i)T + (-0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 + (-0.984 + 0.173i)T^{2} \) |
| 13 | \( 1 + (-1.28 + 0.597i)T + (0.642 - 0.766i)T^{2} \) |
| 17 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 19 | \( 1 + (0.223 + 0.266i)T + (-0.173 + 0.984i)T^{2} \) |
| 23 | \( 1 + (0.173 + 0.984i)T^{2} \) |
| 29 | \( 1 + (0.984 + 0.173i)T^{2} \) |
| 31 | \( 1 + (-1.48 - 1.03i)T + (0.342 + 0.939i)T^{2} \) |
| 37 | \( 1 + (1.32 + 1.11i)T + (0.173 + 0.984i)T^{2} \) |
| 41 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 43 | \( 1 + (1.58 + 0.424i)T + (0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 + (0.642 + 0.766i)T^{2} \) |
| 53 | \( 1 + (-0.984 - 0.173i)T^{2} \) |
| 59 | \( 1 + (0.642 - 0.766i)T^{2} \) |
| 61 | \( 1 + (0.233 + 0.642i)T + (-0.766 + 0.642i)T^{2} \) |
| 67 | \( 1 + (-0.439 + 1.20i)T + (-0.766 - 0.642i)T^{2} \) |
| 71 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 79 | \( 1 + (0.642 - 1.76i)T + (-0.766 - 0.642i)T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 97 | \( 1 + (0.592 + 0.342i)T + (0.5 + 0.866i)T^{2} \) |
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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
−10.11343249343541462199269267416, −9.301957642527141087249358528858, −8.515242543502443918124093847046, −8.249826599457987182356055062699, −6.83116252196120273159809240445, −6.12167348303368771039523245804, −5.26487420300633357771900571856, −3.64212797802653477319939507111, −2.85713821964096419438579830728, −1.80949748560620807042071841489,
1.52549644750459409713258768050, 3.22237672484143143401125251078, 3.95020842712404941683038229021, 4.59266172651457502433097959950, 6.22362363066231412416589952140, 7.00816834190973115282769354381, 7.934453507524814287270604969800, 8.582975870235400204792974471333, 9.754998969053043616938928918240, 10.14847663027471923878196318696