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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.630172600\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.630172600\) |
\(L(\frac{3}{2})\) |
|
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
−8.806521443817236436082391013192, −8.058021560883418098093678656701, −7.22372983546029733968744825466, −6.49136920534852276201503134624, −5.64093376788795489009802645719, −4.97841640180916414662446311510, −4.13534507678828423403722059559, −3.33212284215746924064633671685, −1.89391777779346160987247761598, −1.36884072233065250071738913523,
0.47793955577332570900640088161, 1.94343588405441942659559806328, 2.75940405472030757182818077422, 3.73300437114305883411929180311, 4.42682838833444422738022997011, 5.65975145181356188248389810486, 6.29996019405270781317064729077, 6.65361833739716446526175153808, 7.71420777938584150005123227260, 8.523206547480919326328770445403