L(s) = 1 | + 3.69·5-s + (−2.41 − 1.08i)7-s − 3.41i·11-s + 5.22i·13-s + 6.75·17-s + 2.16i·19-s + 6.24i·23-s + 8.65·25-s − 2.58i·29-s + 10.4i·31-s + (−8.92 − 4i)35-s − 4·37-s − 0.634·41-s − 6.48·43-s + 3.06·47-s + ⋯ |
L(s) = 1 | + 1.65·5-s + (−0.912 − 0.409i)7-s − 1.02i·11-s + 1.44i·13-s + 1.63·17-s + 0.496i·19-s + 1.30i·23-s + 1.73·25-s − 0.480i·29-s + 1.87i·31-s + (−1.50 − 0.676i)35-s − 0.657·37-s − 0.0990·41-s − 0.988·43-s + 0.446·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 - 0.508i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.860 - 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.450997546\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.450997546\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.41 + 1.08i)T \) |
good | 5 | \( 1 - 3.69T + 5T^{2} \) |
| 11 | \( 1 + 3.41iT - 11T^{2} \) |
| 13 | \( 1 - 5.22iT - 13T^{2} \) |
| 17 | \( 1 - 6.75T + 17T^{2} \) |
| 19 | \( 1 - 2.16iT - 19T^{2} \) |
| 23 | \( 1 - 6.24iT - 23T^{2} \) |
| 29 | \( 1 + 2.58iT - 29T^{2} \) |
| 31 | \( 1 - 10.4iT - 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + 0.634T + 41T^{2} \) |
| 43 | \( 1 + 6.48T + 43T^{2} \) |
| 47 | \( 1 - 3.06T + 47T^{2} \) |
| 53 | \( 1 - 2.58iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 1.65T + 67T^{2} \) |
| 71 | \( 1 + 7.41iT - 71T^{2} \) |
| 73 | \( 1 - 0.896iT - 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 - 6.75T + 89T^{2} \) |
| 97 | \( 1 + 9.55iT - 97T^{2} \) |
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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
−8.782131712437457196965616686114, −7.69470215550602284485393310974, −6.82325371749109966642849088455, −6.25925566945141184606031580766, −5.66550871920289799192745205003, −4.99296026460433578771458108797, −3.62013692312479704871416979527, −3.16735598379207830099146509514, −1.91019926360896291257226374296, −1.14999303700002250799834961115,
0.76160539689810389705420766129, 2.07098173943504441442458408217, 2.70744982575330537515706450217, 3.55570125385755950123583228550, 4.93221825036720894345856719773, 5.48205761348377958117141945460, 6.09760958351222761442300380118, 6.74067694689235266940850776823, 7.61430474158881215242926851013, 8.459083115935112536198204548939