| L(s) = 1 | + 3·5-s − 7-s − 3·11-s + 4·13-s − 2·19-s + 6·23-s + 4·25-s + 6·29-s + 5·31-s − 3·35-s − 2·37-s + 6·41-s + 10·43-s − 6·47-s − 6·49-s + 9·53-s − 9·55-s + 12·59-s − 8·61-s + 12·65-s − 14·67-s − 7·73-s + 3·77-s + 8·79-s − 3·83-s + 18·89-s − 4·91-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 0.377·7-s − 0.904·11-s + 1.10·13-s − 0.458·19-s + 1.25·23-s + 4/5·25-s + 1.11·29-s + 0.898·31-s − 0.507·35-s − 0.328·37-s + 0.937·41-s + 1.52·43-s − 0.875·47-s − 6/7·49-s + 1.23·53-s − 1.21·55-s + 1.56·59-s − 1.02·61-s + 1.48·65-s − 1.71·67-s − 0.819·73-s + 0.341·77-s + 0.900·79-s − 0.329·83-s + 1.90·89-s − 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.186066964\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.186066964\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−9.254274842104821367212382551306, −8.728358107276488427256715933065, −7.78966227402229455461895642068, −6.69450931342145073952637354839, −6.11240325026052414300507082707, −5.39588744112665066773671436635, −4.45861885513123914392518249053, −3.13246179353932073023232514138, −2.34838622997525881980959331572, −1.07855163027675466308171882838,
1.07855163027675466308171882838, 2.34838622997525881980959331572, 3.13246179353932073023232514138, 4.45861885513123914392518249053, 5.39588744112665066773671436635, 6.11240325026052414300507082707, 6.69450931342145073952637354839, 7.78966227402229455461895642068, 8.728358107276488427256715933065, 9.254274842104821367212382551306
