| L(s) = 1 | + (−1.16 − 2.37i)7-s − 3.74i·11-s + 0.841i·13-s − 3.36·17-s − 4.55i·19-s − 7.64i·23-s + 1.41i·29-s − 0.979i·31-s − 2.32·37-s + 10.3·41-s + 10.8·43-s − 7.91·47-s + (−4.29 + 5.53i)49-s + 4.35i·53-s + 1.38·59-s + ⋯ |
| L(s) = 1 | + (−0.439 − 0.898i)7-s − 1.12i·11-s + 0.233i·13-s − 0.814·17-s − 1.04i·19-s − 1.59i·23-s + 0.262i·29-s − 0.175i·31-s − 0.382·37-s + 1.61·41-s + 1.64·43-s − 1.15·47-s + (−0.613 + 0.790i)49-s + 0.598i·53-s + 0.180·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.159i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 - 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7023540179\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7023540179\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.16 + 2.37i)T \) |
| good | 11 | \( 1 + 3.74iT - 11T^{2} \) |
| 13 | \( 1 - 0.841iT - 13T^{2} \) |
| 17 | \( 1 + 3.36T + 17T^{2} \) |
| 19 | \( 1 + 4.55iT - 19T^{2} \) |
| 23 | \( 1 + 7.64iT - 23T^{2} \) |
| 29 | \( 1 - 1.41iT - 29T^{2} \) |
| 31 | \( 1 + 0.979iT - 31T^{2} \) |
| 37 | \( 1 + 2.32T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 + 7.91T + 47T^{2} \) |
| 53 | \( 1 - 4.35iT - 53T^{2} \) |
| 59 | \( 1 - 1.38T + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 + 3.74iT - 71T^{2} \) |
| 73 | \( 1 - 8.66iT - 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 + 3.14T + 83T^{2} \) |
| 89 | \( 1 + 3.91T + 89T^{2} \) |
| 97 | \( 1 - 14.8iT - 97T^{2} \) |
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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
−7.59302891629683785581578340114, −6.90192866211734723661193859958, −6.35674443541126089585769590394, −5.65992880138355157717962044643, −4.51117964388255489338546203194, −4.18873714475945728480512356160, −3.09688485717602203939922494031, −2.47958023238724219730623863888, −1.05827601605462802064401854481, −0.18983181221468286730514107619,
1.52649820728989347473727797315, 2.29365519470173742202023857070, 3.17548955302980455888315968704, 4.06525378680529839722491267163, 4.82262195159668725777453190702, 5.74185171443403369713128614452, 6.09744962811822985710426560089, 7.14249712065559846177027922626, 7.57969661198272227201178860802, 8.453540187353376052753062282680
