L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.866 − 0.5i)5-s − 0.999i·8-s + (0.499 + 0.866i)10-s + (−1.5 + 0.866i)13-s + (−0.5 + 0.866i)16-s + i·17-s − 0.999i·20-s + (0.499 + 0.866i)25-s + 1.73·26-s + (−0.866 + 1.5i)29-s + (0.866 − 0.499i)32-s + (0.5 − 0.866i)34-s − 1.73i·37-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.866 − 0.5i)5-s − 0.999i·8-s + (0.499 + 0.866i)10-s + (−1.5 + 0.866i)13-s + (−0.5 + 0.866i)16-s + i·17-s − 0.999i·20-s + (0.499 + 0.866i)25-s + 1.73·26-s + (−0.866 + 1.5i)29-s + (0.866 − 0.499i)32-s + (0.5 − 0.866i)34-s − 1.73i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3358156328\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3358156328\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.866 + 0.5i)T \) |
good | 7 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 - iT - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + 1.73iT - T^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 - 2iT - T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 1.73iT - T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + 1.73T + T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T^{2} \) |
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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
−9.551560088877785304326659500571, −9.054620009933562452144958124251, −8.325253677729579159731655156240, −7.34576179490383095842049067331, −7.12821107160183185585893551568, −5.71290930797732832286846249316, −4.48801351537193899049276887311, −3.83415364223933840631437076019, −2.64503752200081219583980833089, −1.47878946688837208851053819259,
0.34606435829644916010261749344, 2.27115224643207488907508161310, 3.18975803668041914658563951657, 4.62597409388878188053652233171, 5.37077426520007940626515036357, 6.49725292776363356088912522476, 7.21738693550433556715148964045, 7.80475590398729927718444856660, 8.376593843462323559054166952062, 9.560586721987748224618211223301