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
−9.398661319432373077377460821290, −8.715400387625001276348009738225, −7.86163965186582158886816316506, −7.42587666762991906151973686657, −5.98822208701618526537470619763, −5.60472589354089515552422742235, −4.78800305526582150374618692628, −3.62834184681690635374699166656, −3.14286806131684989039542006484, −2.13322146397854402876433217986,
0.38505450821750000300779799779, 1.69968036495218846029467296794, 2.55109123797607687912030611546, 3.99158741392798706638315752250, 4.72725168396198014012369131786, 5.34305900732435903802360748982, 6.59640206573019765371020053732, 7.28942498113989425927474012080, 7.66657356528976352281762309554, 8.495624938794389860622123673557