| L(s) = 1 | − 3-s + 5-s + 2·7-s + 9-s + 2·11-s − 15-s − 2·17-s − 2·21-s + 8·23-s + 25-s − 27-s − 2·33-s + 2·35-s − 4·37-s + 8·41-s + 6·43-s + 45-s + 8·47-s − 3·49-s + 2·51-s + 10·53-s + 2·55-s − 8·59-s + 2·61-s + 2·63-s − 8·69-s + 8·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.603·11-s − 0.258·15-s − 0.485·17-s − 0.436·21-s + 1.66·23-s + 1/5·25-s − 0.192·27-s − 0.348·33-s + 0.338·35-s − 0.657·37-s + 1.24·41-s + 0.914·43-s + 0.149·45-s + 1.16·47-s − 3/7·49-s + 0.280·51-s + 1.37·53-s + 0.269·55-s − 1.04·59-s + 0.256·61-s + 0.251·63-s − 0.963·69-s + 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.135217616\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.135217616\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−13.85031261825658, −13.50914063993513, −12.74823397511512, −12.52823298514621, −11.89084130334117, −11.31215268084585, −10.98799879381884, −10.61973228189606, −9.955422663376919, −9.341517521931825, −8.931344365950799, −8.522865415353457, −7.711914562659656, −7.197680011415542, −6.818994381253337, −6.121738773808268, −5.672731052292289, −5.100874237189009, −4.573057828517795, −4.103501692555026, −3.316954735587668, −2.552199912271409, −1.936605950081641, −1.183207142349893, −0.6639299896150757,
0.6639299896150757, 1.183207142349893, 1.936605950081641, 2.552199912271409, 3.316954735587668, 4.103501692555026, 4.573057828517795, 5.100874237189009, 5.672731052292289, 6.121738773808268, 6.818994381253337, 7.197680011415542, 7.711914562659656, 8.522865415353457, 8.931344365950799, 9.341517521931825, 9.955422663376919, 10.61973228189606, 10.98799879381884, 11.31215268084585, 11.89084130334117, 12.52823298514621, 12.74823397511512, 13.50914063993513, 13.85031261825658
