| L(s) = 1 | + 2.99·3-s + 0.369·5-s − 7-s + 5.96·9-s + 3.93·11-s + 6.07·13-s + 1.10·15-s + 6.75·17-s + 0.117·19-s − 2.99·21-s − 8.67·23-s − 4.86·25-s + 8.86·27-s − 6.27·29-s + 0.808·31-s + 11.7·33-s − 0.369·35-s + 10.1·37-s + 18.1·39-s − 7.27·41-s − 0.0365·43-s + 2.20·45-s − 4.06·47-s + 49-s + 20.2·51-s + 2.53·53-s + 1.45·55-s + ⋯ |
| L(s) = 1 | + 1.72·3-s + 0.165·5-s − 0.377·7-s + 1.98·9-s + 1.18·11-s + 1.68·13-s + 0.285·15-s + 1.63·17-s + 0.0268·19-s − 0.653·21-s − 1.80·23-s − 0.972·25-s + 1.70·27-s − 1.16·29-s + 0.145·31-s + 2.05·33-s − 0.0625·35-s + 1.67·37-s + 2.91·39-s − 1.13·41-s − 0.00557·43-s + 0.328·45-s − 0.593·47-s + 0.142·49-s + 2.83·51-s + 0.348·53-s + 0.196·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.292254459\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.292254459\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 - 2.99T + 3T^{2} \) |
| 5 | \( 1 - 0.369T + 5T^{2} \) |
| 11 | \( 1 - 3.93T + 11T^{2} \) |
| 13 | \( 1 - 6.07T + 13T^{2} \) |
| 17 | \( 1 - 6.75T + 17T^{2} \) |
| 19 | \( 1 - 0.117T + 19T^{2} \) |
| 23 | \( 1 + 8.67T + 23T^{2} \) |
| 29 | \( 1 + 6.27T + 29T^{2} \) |
| 31 | \( 1 - 0.808T + 31T^{2} \) |
| 37 | \( 1 - 10.1T + 37T^{2} \) |
| 41 | \( 1 + 7.27T + 41T^{2} \) |
| 43 | \( 1 + 0.0365T + 43T^{2} \) |
| 47 | \( 1 + 4.06T + 47T^{2} \) |
| 53 | \( 1 - 2.53T + 53T^{2} \) |
| 59 | \( 1 + 5.53T + 59T^{2} \) |
| 61 | \( 1 + 6.02T + 61T^{2} \) |
| 67 | \( 1 - 4.88T + 67T^{2} \) |
| 71 | \( 1 + 2.35T + 71T^{2} \) |
| 73 | \( 1 + 6.82T + 73T^{2} \) |
| 79 | \( 1 + 6.35T + 79T^{2} \) |
| 83 | \( 1 - 8.65T + 83T^{2} \) |
| 89 | \( 1 + 3.56T + 89T^{2} \) |
| 97 | \( 1 + 0.211T + 97T^{2} \) |
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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
−8.502251722055035296375099891210, −7.977118637644660131640364526767, −7.37320152121150843351000079599, −6.24415882367781054513801498722, −5.85173458997924305422619556578, −4.25273204138558546112470314257, −3.66304520923902257056926596845, −3.26094741349623082703527137978, −1.98535265001311835568603038253, −1.29927155553044815731985462097,
1.29927155553044815731985462097, 1.98535265001311835568603038253, 3.26094741349623082703527137978, 3.66304520923902257056926596845, 4.25273204138558546112470314257, 5.85173458997924305422619556578, 6.24415882367781054513801498722, 7.37320152121150843351000079599, 7.977118637644660131640364526767, 8.502251722055035296375099891210
