L(s) = 1 | + (0.939 + 1.62i)2-s + (−1.26 + 2.19i)4-s + (−0.5 + 0.866i)5-s − 2.87·8-s − 1.87·10-s + (−1.43 − 2.49i)16-s + 0.347·17-s + 0.347·19-s + (−1.26 − 2.19i)20-s + (−0.766 + 1.32i)23-s + (−0.499 − 0.866i)25-s + (0.939 − 1.62i)31-s + (1.26 − 2.19i)32-s + (0.326 + 0.565i)34-s + (0.326 + 0.565i)38-s + ⋯ |
L(s) = 1 | + (0.939 + 1.62i)2-s + (−1.26 + 2.19i)4-s + (−0.5 + 0.866i)5-s − 2.87·8-s − 1.87·10-s + (−1.43 − 2.49i)16-s + 0.347·17-s + 0.347·19-s + (−1.26 − 2.19i)20-s + (−0.766 + 1.32i)23-s + (−0.499 − 0.866i)25-s + (0.939 − 1.62i)31-s + (1.26 − 2.19i)32-s + (0.326 + 0.565i)34-s + (0.326 + 0.565i)38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1215 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1215 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.332133936\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.332133936\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
good | 2 | \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 - 0.347T + T^{2} \) |
| 19 | \( 1 - 0.347T + T^{2} \) |
| 23 | \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 - 1.53T + T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \) |
| 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
−10.24114248022359554978059938147, −9.331575633936871911714801604332, −8.248255734400945144971980599183, −7.62738967628092990669454838971, −7.13041826432147015770932999067, −6.11992116875315679826834272891, −5.64514839872994527615271212167, −4.41897988616347564626738899807, −3.76125756602634345653048913374, −2.77737061251395926004580582686,
0.915388144085387702778319729450, 2.15744692020729398935865247413, 3.31461914421157489832965251043, 4.12743097976949670739868866370, 4.91064720073111717441306637375, 5.57619555562649377205914239020, 6.76519808403005850457485730661, 8.231430168899691163926814612751, 8.845421706859847399567282464976, 9.867522852847059923464483018728