L(s) = 1 | + (1.22 + 0.707i)2-s + (0.5 + 0.866i)3-s + (0.499 + 0.866i)4-s + (0.5 + 0.866i)5-s + 1.41i·6-s + (−1.22 − 0.707i)7-s + (−0.499 + 0.866i)9-s + 1.41i·10-s + (−0.5 + 0.866i)12-s + (−0.999 − 1.73i)14-s + (−0.499 + 0.866i)15-s + (0.499 − 0.866i)16-s + (−1.22 + 0.707i)18-s + (−0.5 + 0.866i)20-s − 1.41i·21-s + ⋯ |
L(s) = 1 | + (1.22 + 0.707i)2-s + (0.5 + 0.866i)3-s + (0.499 + 0.866i)4-s + (0.5 + 0.866i)5-s + 1.41i·6-s + (−1.22 − 0.707i)7-s + (−0.499 + 0.866i)9-s + 1.41i·10-s + (−0.5 + 0.866i)12-s + (−0.999 − 1.73i)14-s + (−0.499 + 0.866i)15-s + (0.499 − 0.866i)16-s + (−1.22 + 0.707i)18-s + (−0.5 + 0.866i)20-s − 1.41i·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.262 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.262 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.103875657\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.103875657\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 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 | \( 1 - T + T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 - 1.41iT - T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−10.17277726283853219649859807951, −9.748844464190782596013385895739, −8.688234422047237810482274753567, −7.41655155789646222772413484922, −6.76905103510706204498182617591, −6.05841395278505951749810063999, −5.14571203954136845081634654854, −4.10667743345344212108250949011, −3.42175591788946648275141168233, −2.66182498676302749541908147473,
1.53911232748449003147121250377, 2.65769916299205601557110762508, 3.29622586802518816694758198894, 4.50045355799031918187645100809, 5.60857862348732504522626856422, 6.10160206583250533886459593606, 7.10297103897251148556946399910, 8.415217593764334968797809938997, 8.966222470897175577346281033624, 9.770163954859646592040401083465