| L(s) = 1 | + 4·5-s − 6·11-s + 2·13-s + 4·17-s − 19-s + 6·23-s + 11·25-s + 10·29-s + 8·31-s − 10·37-s − 6·41-s − 4·43-s + 6·47-s − 7·49-s + 2·53-s − 24·55-s + 4·59-s + 10·61-s + 8·65-s − 12·67-s + 12·71-s − 6·73-s − 4·79-s + 14·83-s + 16·85-s − 6·89-s − 4·95-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 1.80·11-s + 0.554·13-s + 0.970·17-s − 0.229·19-s + 1.25·23-s + 11/5·25-s + 1.85·29-s + 1.43·31-s − 1.64·37-s − 0.937·41-s − 0.609·43-s + 0.875·47-s − 49-s + 0.274·53-s − 3.23·55-s + 0.520·59-s + 1.28·61-s + 0.992·65-s − 1.46·67-s + 1.42·71-s − 0.702·73-s − 0.450·79-s + 1.53·83-s + 1.73·85-s − 0.635·89-s − 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.288600590\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.288600590\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−9.843187490189309532847501765164, −8.733630445014506961411164280575, −8.188739896970630999991719296997, −6.96144809894394358067044941513, −6.24670329210725230269790841808, −5.29073756521375423883662858784, −4.96968245486932091706458917661, −3.13358913525565115720185038251, −2.44562128151779883150829106452, −1.19815847963162522724198031958,
1.19815847963162522724198031958, 2.44562128151779883150829106452, 3.13358913525565115720185038251, 4.96968245486932091706458917661, 5.29073756521375423883662858784, 6.24670329210725230269790841808, 6.96144809894394358067044941513, 8.188739896970630999991719296997, 8.733630445014506961411164280575, 9.843187490189309532847501765164
