L(s) = 1 | + (0.939 + 1.62i)2-s + (−0.5 + 0.866i)3-s + (−1.26 + 2.19i)4-s + (−0.173 + 0.300i)5-s − 1.87·6-s − 2.87·8-s + (−0.499 − 0.866i)9-s − 0.652·10-s + (−0.173 − 0.300i)11-s + (−1.26 − 2.19i)12-s + (−0.173 − 0.300i)15-s + (−1.43 − 2.49i)16-s − 1.87·17-s + (0.939 − 1.62i)18-s + (−0.439 − 0.761i)20-s + ⋯ |
L(s) = 1 | + (0.939 + 1.62i)2-s + (−0.5 + 0.866i)3-s + (−1.26 + 2.19i)4-s + (−0.173 + 0.300i)5-s − 1.87·6-s − 2.87·8-s + (−0.499 − 0.866i)9-s − 0.652·10-s + (−0.173 − 0.300i)11-s + (−1.26 − 2.19i)12-s + (−0.173 − 0.300i)15-s + (−1.43 − 2.49i)16-s − 1.87·17-s + (0.939 − 1.62i)18-s + (−0.439 − 0.761i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8885387204\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8885387204\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 - 0.866i)T \) |
| 239 | \( 1 + (0.5 - 0.866i)T \) |
good | 2 | \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + 1.87T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 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 | \( 1 - T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - 2T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 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
−9.533888194589612177974357397157, −8.835634191427802305874090182157, −8.284401951653896459422364074392, −7.08928231482185620045690041881, −6.74056840997520500573970417614, −5.84369756202022242896691983830, −5.20225252725517574738031374859, −4.44879429440895800855772022491, −3.78626034075234929663284957098, −2.86253975413918395600563704371,
0.46456603179146260202630408286, 1.90619037866651601221210694555, 2.38668220169123174547162429190, 3.64457234661211147124053425892, 4.61893615851963534583311775643, 5.10563675585137393518917705686, 6.12350796761916419612689115212, 6.78742262621421082016946782281, 7.992371777479416445441818819419, 8.905093284009209226299050422708