L(s) = 1 | + (−0.142 − 0.989i)2-s + (0.415 − 0.909i)3-s + (−0.959 + 0.281i)4-s + (−0.959 − 0.281i)6-s + (0.841 − 0.540i)7-s + (0.415 + 0.909i)8-s + (−0.654 − 0.755i)9-s + (−0.142 + 0.989i)12-s + (−0.239 − 0.153i)13-s + (−0.654 − 0.755i)14-s + (0.841 − 0.540i)16-s + (1.25 + 0.368i)17-s + (−0.654 + 0.755i)18-s + (−0.142 − 0.989i)21-s + (0.415 + 0.909i)23-s + 24-s + ⋯ |
L(s) = 1 | + (−0.142 − 0.989i)2-s + (0.415 − 0.909i)3-s + (−0.959 + 0.281i)4-s + (−0.959 − 0.281i)6-s + (0.841 − 0.540i)7-s + (0.415 + 0.909i)8-s + (−0.654 − 0.755i)9-s + (−0.142 + 0.989i)12-s + (−0.239 − 0.153i)13-s + (−0.654 − 0.755i)14-s + (0.841 − 0.540i)16-s + (1.25 + 0.368i)17-s + (−0.654 + 0.755i)18-s + (−0.142 − 0.989i)21-s + (0.415 + 0.909i)23-s + 24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 + 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 + 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.304043277\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.304043277\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.142 + 0.989i)T \) |
| 3 | \( 1 + (-0.415 + 0.909i)T \) |
| 7 | \( 1 + (-0.841 + 0.540i)T \) |
| 23 | \( 1 + (-0.415 - 0.909i)T \) |
good | 5 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 11 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 13 | \( 1 + (0.239 + 0.153i)T + (0.415 + 0.909i)T^{2} \) |
| 17 | \( 1 + (-1.25 - 0.368i)T + (0.841 + 0.540i)T^{2} \) |
| 19 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 29 | \( 1 + (0.797 + 0.234i)T + (0.841 + 0.540i)T^{2} \) |
| 31 | \( 1 + (0.797 + 1.74i)T + (-0.654 + 0.755i)T^{2} \) |
| 37 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 41 | \( 1 + (-0.857 + 0.989i)T + (-0.142 - 0.989i)T^{2} \) |
| 43 | \( 1 + (-0.345 + 0.755i)T + (-0.654 - 0.755i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (1.61 - 1.03i)T + (0.415 - 0.909i)T^{2} \) |
| 59 | \( 1 + (-1.41 - 0.909i)T + (0.415 + 0.909i)T^{2} \) |
| 61 | \( 1 + (-0.698 - 1.53i)T + (-0.654 + 0.755i)T^{2} \) |
| 67 | \( 1 + (0.239 + 1.66i)T + (-0.959 + 0.281i)T^{2} \) |
| 71 | \( 1 + (0.239 + 1.66i)T + (-0.959 + 0.281i)T^{2} \) |
| 73 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 79 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 83 | \( 1 + (-0.186 - 0.215i)T + (-0.142 + 0.989i)T^{2} \) |
| 89 | \( 1 + (0.797 - 1.74i)T + (-0.654 - 0.755i)T^{2} \) |
| 97 | \( 1 + (0.142 + 0.989i)T^{2} \) |
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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
−8.210638112395416750796427337659, −7.69747858789301329995914195579, −7.32019469293215066872112008456, −5.92082615962625663171698482711, −5.35426302118311827831216772755, −4.15748757804780629126158682740, −3.59121209857254274594984890645, −2.51244651293752471581004375455, −1.75557322518136941942353173081, −0.801418905165311988602109549761,
1.52980698144517734094599600492, 2.92556541661257631657899978343, 3.77471035409555323613465882848, 4.78017952511071122202262444479, 5.19720355625188535470127332405, 5.81319632696536201407002571325, 6.94916127478201778953726572385, 7.66180207047175466060477441118, 8.322518729449271066088288369909, 8.842143675082635245820296018612