L(s) = 1 | − i·3-s + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)7-s − 9-s + (−0.5 + 0.866i)11-s + (−0.866 + 0.5i)13-s + (−0.866 + 0.5i)15-s − i·19-s + (−0.866 − 0.5i)21-s + (−0.5 − 0.866i)23-s + i·27-s + (−0.866 − 0.5i)29-s + (−0.866 + 0.5i)31-s + (0.866 + 0.5i)33-s − 0.999·35-s + ⋯ |
L(s) = 1 | − i·3-s + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)7-s − 9-s + (−0.5 + 0.866i)11-s + (−0.866 + 0.5i)13-s + (−0.866 + 0.5i)15-s − i·19-s + (−0.866 − 0.5i)21-s + (−0.5 − 0.866i)23-s + i·27-s + (−0.866 − 0.5i)29-s + (−0.866 + 0.5i)31-s + (0.866 + 0.5i)33-s − 0.999·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6380062153\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6380062153\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 19 | \( 1 + iT \) |
good | 5 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
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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
−8.495624938794389860622123673557, −7.66657356528976352281762309554, −7.28942498113989425927474012080, −6.59640206573019765371020053732, −5.34305900732435903802360748982, −4.72725168396198014012369131786, −3.99158741392798706638315752250, −2.55109123797607687912030611546, −1.69968036495218846029467296794, −0.38505450821750000300779799779,
2.13322146397854402876433217986, 3.14286806131684989039542006484, 3.62834184681690635374699166656, 4.78800305526582150374618692628, 5.60472589354089515552422742235, 5.98822208701618526537470619763, 7.42587666762991906151973686657, 7.86163965186582158886816316506, 8.715400387625001276348009738225, 9.398661319432373077377460821290