L(s) = 1 | + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)9-s + (1.5 − 0.866i)11-s + 1.73i·17-s + 19-s + (−0.5 − 0.866i)25-s + 0.999·27-s + (−1.5 − 0.866i)33-s + (1.5 + 0.866i)41-s + (−0.5 − 0.866i)43-s + (0.5 − 0.866i)49-s + (1.49 − 0.866i)51-s + (−0.5 − 0.866i)57-s + (−1.5 − 0.866i)59-s + (0.5 − 0.866i)67-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)9-s + (1.5 − 0.866i)11-s + 1.73i·17-s + 19-s + (−0.5 − 0.866i)25-s + 0.999·27-s + (−1.5 − 0.866i)33-s + (1.5 + 0.866i)41-s + (−0.5 − 0.866i)43-s + (0.5 − 0.866i)49-s + (1.49 − 0.866i)51-s + (−0.5 − 0.866i)57-s + (−1.5 − 0.866i)59-s + (0.5 − 0.866i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.091106174\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.091106174\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
good | 5 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 - 1.73iT - T^{2} \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T + T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
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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
−8.920864652221798222613614451981, −8.239751985672739571394955584533, −7.57894476478838992490754431944, −6.46917121478548673741079859262, −6.23466296894658396827754149097, −5.37353751430660881773428150945, −4.18556991366505457510458228159, −3.34970570359753199152917613784, −1.99298012122772766376033832948, −1.03550590854060817784974691191,
1.18763706713086880762617723819, 2.76516435066905591291325582101, 3.76271415065123657578983690301, 4.50678680505112285639292184374, 5.25708792688710431113692871603, 6.08806119553662911667452362845, 7.00039570109798826624446317857, 7.56296797001068176148630041079, 8.922326555052859996757552623511, 9.521802796018740784390201812162