L(s) = 1 | + (1 + i)5-s + (5 − 5i)13-s − 8·17-s − 3i·25-s + (−3 + 3i)29-s + (−7 − 7i)37-s − 8i·41-s + 7·49-s + (−9 − 9i)53-s + (−11 + 11i)61-s + 10·65-s − 6i·73-s + (−8 − 8i)85-s + 10i·89-s − 8·97-s + ⋯ |
L(s) = 1 | + (0.447 + 0.447i)5-s + (1.38 − 1.38i)13-s − 1.94·17-s − 0.600i·25-s + (−0.557 + 0.557i)29-s + (−1.15 − 1.15i)37-s − 1.24i·41-s + 49-s + (−1.23 − 1.23i)53-s + (−1.40 + 1.40i)61-s + 1.24·65-s − 0.702i·73-s + (−0.867 − 0.867i)85-s + 1.05i·89-s − 0.812·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.167334562\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.167334562\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-1 - i)T + 5iT^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 11iT^{2} \) |
| 13 | \( 1 + (-5 + 5i)T - 13iT^{2} \) |
| 17 | \( 1 + 8T + 17T^{2} \) |
| 19 | \( 1 - 19iT^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + (3 - 3i)T - 29iT^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + (7 + 7i)T + 37iT^{2} \) |
| 41 | \( 1 + 8iT - 41T^{2} \) |
| 43 | \( 1 + 43iT^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + (9 + 9i)T + 53iT^{2} \) |
| 59 | \( 1 + 59iT^{2} \) |
| 61 | \( 1 + (11 - 11i)T - 61iT^{2} \) |
| 67 | \( 1 - 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 83iT^{2} \) |
| 89 | \( 1 - 10iT - 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−8.220522536698658480646462780140, −7.26197015724347531473370765375, −6.61349699965458914484771177815, −5.91919397674936643478558748056, −5.31440857668138355638308606480, −4.24017126961895235332563295210, −3.49159543658126018020852368110, −2.58452947807804368614250211363, −1.70232929808943292517189370194, −0.30233906455247536073770664271,
1.38799752323622777939211621540, 2.00672824983801974360766574124, 3.21067521998966670090873657728, 4.22540038791837898455467945269, 4.66927461235276884505112560817, 5.72367634540543803081488787697, 6.44556821509289148266109303959, 6.85764210818605317440192271871, 7.943195396154858358590897582081, 8.748955858567181465749111322743