| L(s) = 1 | + 3-s + 2·5-s + 7-s + 9-s + 3·11-s + 4·13-s + 2·15-s − 17-s − 6·19-s + 21-s + 4·23-s − 25-s + 27-s + 5·29-s + 5·31-s + 3·33-s + 2·35-s − 11·37-s + 4·39-s + 43-s + 2·45-s + 9·47-s + 49-s − 51-s − 6·53-s + 6·55-s − 6·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s + 0.904·11-s + 1.10·13-s + 0.516·15-s − 0.242·17-s − 1.37·19-s + 0.218·21-s + 0.834·23-s − 1/5·25-s + 0.192·27-s + 0.928·29-s + 0.898·31-s + 0.522·33-s + 0.338·35-s − 1.80·37-s + 0.640·39-s + 0.152·43-s + 0.298·45-s + 1.31·47-s + 1/7·49-s − 0.140·51-s − 0.824·53-s + 0.809·55-s − 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141204 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141204 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.544726004\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.544726004\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 41 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.57487780776182, −13.08892288590880, −12.36458762542454, −12.17165229204847, −11.41379121377146, −10.93242986338158, −10.32431204388117, −10.28044870944760, −9.299706039176975, −8.923828740763053, −8.822246735553249, −8.100749509526764, −7.654379870113609, −6.801539466063132, −6.474003308592778, −6.142757436292315, −5.452564914613466, −4.739288035713349, −4.359513072403833, −3.664124903030587, −3.206689083362062, −2.392793035752890, −1.901094191460928, −1.383367992389752, −0.6855121214078532,
0.6855121214078532, 1.383367992389752, 1.901094191460928, 2.392793035752890, 3.206689083362062, 3.664124903030587, 4.359513072403833, 4.739288035713349, 5.452564914613466, 6.142757436292315, 6.474003308592778, 6.801539466063132, 7.654379870113609, 8.100749509526764, 8.822246735553249, 8.923828740763053, 9.299706039176975, 10.28044870944760, 10.32431204388117, 10.93242986338158, 11.41379121377146, 12.17165229204847, 12.36458762542454, 13.08892288590880, 13.57487780776182
