| L(s) = 1 | − 2·5-s + 2·7-s + 4·13-s + 2·17-s − 4·19-s + 4·23-s − 25-s − 6·29-s + 2·31-s − 4·35-s + 8·37-s + 2·41-s − 4·43-s + 12·47-s − 3·49-s − 6·53-s − 4·59-s − 8·65-s + 12·67-s + 12·71-s + 6·73-s + 10·79-s + 16·83-s − 4·85-s − 10·89-s + 8·91-s + 8·95-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.755·7-s + 1.10·13-s + 0.485·17-s − 0.917·19-s + 0.834·23-s − 1/5·25-s − 1.11·29-s + 0.359·31-s − 0.676·35-s + 1.31·37-s + 0.312·41-s − 0.609·43-s + 1.75·47-s − 3/7·49-s − 0.824·53-s − 0.520·59-s − 0.992·65-s + 1.46·67-s + 1.42·71-s + 0.702·73-s + 1.12·79-s + 1.75·83-s − 0.433·85-s − 1.05·89-s + 0.838·91-s + 0.820·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.678620176\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.678620176\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−8.881245043824419739677897153393, −8.064896928768304782309586104473, −7.75078328320192586347280578711, −6.70717720500603434376350537614, −5.88791929751498617888946438034, −4.93451449927809914411399509060, −4.09041041707407492058699548419, −3.42497389478170812159021902601, −2.10734881801434152784586446055, −0.862576593176842099681243144024,
0.862576593176842099681243144024, 2.10734881801434152784586446055, 3.42497389478170812159021902601, 4.09041041707407492058699548419, 4.93451449927809914411399509060, 5.88791929751498617888946438034, 6.70717720500603434376350537614, 7.75078328320192586347280578711, 8.064896928768304782309586104473, 8.881245043824419739677897153393
