| L(s) = 1 | + 2.73·3-s + 1.73·5-s + 7-s + 4.46·9-s + 2.73·11-s − 13-s + 4.73·15-s + 2.73·17-s − 1.73·19-s + 2.73·21-s − 0.464·23-s − 2.00·25-s + 3.99·27-s − 6.46·29-s + 0.267·31-s + 7.46·33-s + 1.73·35-s − 3.26·37-s − 2.73·39-s − 10.3·41-s + 12.4·43-s + 7.73·45-s + 5.73·47-s + 49-s + 7.46·51-s − 9.92·53-s + 4.73·55-s + ⋯ |
| L(s) = 1 | + 1.57·3-s + 0.774·5-s + 0.377·7-s + 1.48·9-s + 0.823·11-s − 0.277·13-s + 1.22·15-s + 0.662·17-s − 0.397·19-s + 0.596·21-s − 0.0967·23-s − 0.400·25-s + 0.769·27-s − 1.20·29-s + 0.0481·31-s + 1.29·33-s + 0.292·35-s − 0.537·37-s − 0.437·39-s − 1.62·41-s + 1.90·43-s + 1.15·45-s + 0.836·47-s + 0.142·49-s + 1.04·51-s − 1.36·53-s + 0.638·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.561885425\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.561885425\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 2.73T + 3T^{2} \) |
| 5 | \( 1 - 1.73T + 5T^{2} \) |
| 11 | \( 1 - 2.73T + 11T^{2} \) |
| 17 | \( 1 - 2.73T + 17T^{2} \) |
| 19 | \( 1 + 1.73T + 19T^{2} \) |
| 23 | \( 1 + 0.464T + 23T^{2} \) |
| 29 | \( 1 + 6.46T + 29T^{2} \) |
| 31 | \( 1 - 0.267T + 31T^{2} \) |
| 37 | \( 1 + 3.26T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 - 12.4T + 43T^{2} \) |
| 47 | \( 1 - 5.73T + 47T^{2} \) |
| 53 | \( 1 + 9.92T + 53T^{2} \) |
| 59 | \( 1 + 3.46T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 - 9.73T + 73T^{2} \) |
| 79 | \( 1 + 3.92T + 79T^{2} \) |
| 83 | \( 1 - 6.26T + 83T^{2} \) |
| 89 | \( 1 - 12.2T + 89T^{2} \) |
| 97 | \( 1 + 16.1T + 97T^{2} \) |
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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
−9.362083240992783811413656878509, −8.861303369306761099942829763423, −7.971168401504575478208028055832, −7.35880563192674116453204198665, −6.32974371578406424734133280782, −5.34944189438522713856290564162, −4.16343404093183945236416664431, −3.39242701045279851893851266153, −2.28935906091270938831535498548, −1.56047964317885369310145996726,
1.56047964317885369310145996726, 2.28935906091270938831535498548, 3.39242701045279851893851266153, 4.16343404093183945236416664431, 5.34944189438522713856290564162, 6.32974371578406424734133280782, 7.35880563192674116453204198665, 7.971168401504575478208028055832, 8.861303369306761099942829763423, 9.362083240992783811413656878509
