L(s) = 1 | + (0.965 + 0.258i)3-s + (−0.866 + 0.5i)4-s + (−0.258 + 0.965i)7-s + (0.866 + 0.499i)9-s + (−0.965 + 0.258i)12-s + (−0.707 + 0.707i)13-s + (0.499 − 0.866i)16-s + (0.866 − 1.5i)19-s + (−0.499 + 0.866i)21-s + (0.707 + 0.707i)27-s + (−0.258 − 0.965i)28-s − 36-s + (−0.448 − 1.67i)37-s + (−0.866 + 0.500i)39-s + (0.707 − 0.707i)48-s + (−0.866 − 0.499i)49-s + ⋯ |
L(s) = 1 | + (0.965 + 0.258i)3-s + (−0.866 + 0.5i)4-s + (−0.258 + 0.965i)7-s + (0.866 + 0.499i)9-s + (−0.965 + 0.258i)12-s + (−0.707 + 0.707i)13-s + (0.499 − 0.866i)16-s + (0.866 − 1.5i)19-s + (−0.499 + 0.866i)21-s + (0.707 + 0.707i)27-s + (−0.258 − 0.965i)28-s − 36-s + (−0.448 − 1.67i)37-s + (−0.866 + 0.500i)39-s + (0.707 − 0.707i)48-s + (−0.866 − 0.499i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.547 - 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.547 - 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9799144735\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9799144735\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.965 - 0.258i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.258 - 0.965i)T \) |
good | 2 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 17 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 19 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.448 + 1.67i)T + (-0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-1.67 - 0.448i)T + (0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 79 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
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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
−11.26813279661501972182012595684, −9.897097841714116489567544236523, −9.211766086133598308527243647541, −8.858856363372930905358353385065, −7.78837246017268703227406417814, −6.94754595606155066951798712363, −5.32560035589190743805259897501, −4.50364774584328870421161627358, −3.34047050755471509032231667467, −2.36390133880571560857729700542,
1.34065392843507600280010637163, 3.16921841480485411844525611973, 4.07759666830518613715473251466, 5.17779153657943987353837753776, 6.44853285150850227592927039147, 7.60850285295451075349875379852, 8.160088118592409073752990852155, 9.284380013958618266755744508710, 10.01761731231109408462918490056, 10.43815219824576882193878437889