L(s) = 1 | + (−1 + 1.73i)5-s + (2 + 3.46i)11-s − 2·13-s + (1 + 1.73i)17-s + (2 − 3.46i)19-s + (−4 + 6.92i)23-s + (0.500 + 0.866i)25-s − 6·29-s + (−4 − 6.92i)31-s + (−3 + 5.19i)37-s + 6·41-s + 4·43-s + (−1 − 1.73i)53-s − 7.99·55-s + (2 + 3.46i)59-s + ⋯ |
L(s) = 1 | + (−0.447 + 0.774i)5-s + (0.603 + 1.04i)11-s − 0.554·13-s + (0.242 + 0.420i)17-s + (0.458 − 0.794i)19-s + (−0.834 + 1.44i)23-s + (0.100 + 0.173i)25-s − 1.11·29-s + (−0.718 − 1.24i)31-s + (−0.493 + 0.854i)37-s + 0.937·41-s + 0.609·43-s + (−0.137 − 0.237i)53-s − 1.07·55-s + (0.260 + 0.450i)59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7357217717\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7357217717\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2 - 3.46i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4 - 6.92i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + (4 + 6.92i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3 - 5.19i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1 + 1.73i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + (5 + 8.66i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-4 + 6.92i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−9.204046596487589180186904514203, −7.85512773338797021093032314339, −7.45974219323980043518733786024, −6.90630717232796421988368871476, −5.96531850397358392662028126581, −5.16727025592672306048199659099, −4.14836954244510441850702376959, −3.56105981485770482564089602739, −2.51177880333869842261581254860, −1.54596178224669240159256437825,
0.22571247862950536842039322377, 1.31174783387325350265933982568, 2.57539903730649162076855502659, 3.66051715115320171014303449147, 4.25537204051607515531005631063, 5.24370443692848804349558117651, 5.84162205342835299652013332632, 6.76247297475051832343364194177, 7.59683149763123796779310282940, 8.272553344426047313612471424621