| L(s) = 1 | + 4·5-s + 2·7-s − 11-s − 2·13-s − 2·17-s + 6·19-s + 4·23-s + 11·25-s + 6·29-s − 4·31-s + 8·35-s − 6·37-s + 10·41-s − 6·43-s − 8·47-s − 3·49-s − 4·55-s + 4·59-s − 6·61-s − 8·65-s − 8·67-s − 2·73-s − 2·77-s + 10·79-s + 12·83-s − 8·85-s − 4·91-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 0.755·7-s − 0.301·11-s − 0.554·13-s − 0.485·17-s + 1.37·19-s + 0.834·23-s + 11/5·25-s + 1.11·29-s − 0.718·31-s + 1.35·35-s − 0.986·37-s + 1.56·41-s − 0.914·43-s − 1.16·47-s − 3/7·49-s − 0.539·55-s + 0.520·59-s − 0.768·61-s − 0.992·65-s − 0.977·67-s − 0.234·73-s − 0.227·77-s + 1.12·79-s + 1.31·83-s − 0.867·85-s − 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.597182985\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.597182985\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−9.421762871553342182410427682291, −8.851831118869979972078892187391, −7.80418860825074301985275426770, −6.93284037083089668438228296773, −6.11122341011904446133713007911, −5.16180634474034511746575659100, −4.85856628884129953509261647402, −3.14462078817297289229629667834, −2.21549188624395141058385133404, −1.28234579931623690677841956595,
1.28234579931623690677841956595, 2.21549188624395141058385133404, 3.14462078817297289229629667834, 4.85856628884129953509261647402, 5.16180634474034511746575659100, 6.11122341011904446133713007911, 6.93284037083089668438228296773, 7.80418860825074301985275426770, 8.851831118869979972078892187391, 9.421762871553342182410427682291
