| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.090469692\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.090469692\) |
| \(L(\frac{3}{2})\) |
|
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
−8.885275238011435794894293665343, −8.340435092065256955612697266748, −6.99241980041405149293866414758, −6.47387899339104008227717147348, −5.82410412254114334522628992593, −5.14634186114487729918988465375, −3.98662234684322757661079393036, −3.11238199800224418908476741980, −1.82028192806776351820139775816, −1.29739482557638675877716794635,
1.29739482557638675877716794635, 1.82028192806776351820139775816, 3.11238199800224418908476741980, 3.98662234684322757661079393036, 5.14634186114487729918988465375, 5.82410412254114334522628992593, 6.47387899339104008227717147348, 6.99241980041405149293866414758, 8.340435092065256955612697266748, 8.885275238011435794894293665343
