L(s) = 1 | + (0.415 + 0.909i)2-s + (−1.10 − 0.708i)3-s + (−0.654 + 0.755i)4-s + (−0.142 − 0.989i)5-s + (0.186 − 1.29i)6-s + (0.698 + 1.53i)7-s + (−0.959 − 0.281i)8-s + (0.297 + 0.650i)9-s + (0.841 − 0.540i)10-s + (1.25 − 0.368i)12-s + (−1.10 + 1.27i)14-s + (−0.544 + 1.19i)15-s + (−0.142 − 0.989i)16-s + (−0.468 + 0.540i)18-s + (0.841 + 0.540i)20-s + (0.313 − 2.18i)21-s + ⋯ |
L(s) = 1 | + (0.415 + 0.909i)2-s + (−1.10 − 0.708i)3-s + (−0.654 + 0.755i)4-s + (−0.142 − 0.989i)5-s + (0.186 − 1.29i)6-s + (0.698 + 1.53i)7-s + (−0.959 − 0.281i)8-s + (0.297 + 0.650i)9-s + (0.841 − 0.540i)10-s + (1.25 − 0.368i)12-s + (−1.10 + 1.27i)14-s + (−0.544 + 1.19i)15-s + (−0.142 − 0.989i)16-s + (−0.468 + 0.540i)18-s + (0.841 + 0.540i)20-s + (0.313 − 2.18i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.366 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.366 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8653539358\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8653539358\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.415 - 0.909i)T \) |
| 5 | \( 1 + (0.142 + 0.989i)T \) |
| 67 | \( 1 + (0.654 - 0.755i)T \) |
good | 3 | \( 1 + (1.10 + 0.708i)T + (0.415 + 0.909i)T^{2} \) |
| 7 | \( 1 + (-0.698 - 1.53i)T + (-0.654 + 0.755i)T^{2} \) |
| 11 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 13 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 17 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 19 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 23 | \( 1 + (-1.68 - 1.08i)T + (0.415 + 0.909i)T^{2} \) |
| 29 | \( 1 - 1.68T + T^{2} \) |
| 31 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (-1.25 - 1.45i)T + (-0.142 + 0.989i)T^{2} \) |
| 43 | \( 1 + (-0.857 - 0.989i)T + (-0.142 + 0.989i)T^{2} \) |
| 47 | \( 1 + (1.61 + 1.03i)T + (0.415 + 0.909i)T^{2} \) |
| 53 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 59 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 61 | \( 1 + (-0.0405 + 0.281i)T + (-0.959 - 0.281i)T^{2} \) |
| 71 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 73 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 79 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 83 | \( 1 + (0.118 + 0.822i)T + (-0.959 + 0.281i)T^{2} \) |
| 89 | \( 1 + (-0.698 + 0.449i)T + (0.415 - 0.909i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.596077385806505913481115155240, −8.887322865220425513301343098387, −8.252136736118955263180602547856, −7.45937248653485982060250756084, −6.43513074667473352941654080082, −5.78191531426030016062602703786, −5.11597206240865424442667447942, −4.62876406819905380670006039381, −2.94911662728187002727003522899, −1.28798361802885437212417645902,
0.891176161550498479963551174615, 2.62008969206742194606509624550, 3.77942436831793660399204156343, 4.49527640271813657078879538365, 5.09752358778844220636168990320, 6.25135024805475257236738112461, 6.95837780937214067948199828288, 8.019036523031880890648802454720, 9.246252250816976409704014670040, 10.31818419398216823053333623678