L(s) = 1 | + (−1 + i)5-s − 3i·9-s + (5 + 5i)13-s + 8·17-s + 3i·25-s + (3 + 3i)29-s + (−7 + 7i)37-s − 8i·41-s + (3 + 3i)45-s + 7·49-s + (9 − 9i)53-s + (−11 − 11i)61-s − 10·65-s + 6i·73-s − 9·81-s + ⋯ |
L(s) = 1 | + (−0.447 + 0.447i)5-s − i·9-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 + (0.447 + 0.447i)45-s + 49-s + (1.23 − 1.23i)53-s + (−1.40 − 1.40i)61-s − 1.24·65-s + 0.702i·73-s − 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.40221 + 0.278918i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.40221 + 0.278918i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + 3iT^{2} \) |
| 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
−11.02386975674152690467768652483, −10.08246970809745146308831262477, −9.133028327215641590068972868978, −8.361559591341289211257769191423, −7.17765136673761200507377119936, −6.48900057728029578050946308496, −5.42686253922564345940117789280, −3.89294201188254270664626340478, −3.31615130913095394623605156135, −1.34762916831382432078830694610,
1.09270423426689080323935150257, 2.92793138425852203606671078724, 4.05222104021183707949451473760, 5.30213321231904925464493723223, 5.97668910654611376423357860269, 7.57355083188020345453515884079, 8.048848208304325473870910289010, 8.866597166253576373314699495777, 10.28848396900697739634015199402, 10.59295778641146198552345709771