| L(s) = 1 | − 3-s − 5-s − 7-s − 2·9-s − 4·11-s + 15-s − 17-s + 21-s + 3·23-s + 25-s + 5·27-s + 6·29-s + 4·33-s + 35-s + 11·37-s − 5·41-s + 7·43-s + 2·45-s + 8·47-s + 49-s + 51-s + 3·53-s + 4·55-s + 2·59-s − 8·61-s + 2·63-s + 2·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s − 2/3·9-s − 1.20·11-s + 0.258·15-s − 0.242·17-s + 0.218·21-s + 0.625·23-s + 1/5·25-s + 0.962·27-s + 1.11·29-s + 0.696·33-s + 0.169·35-s + 1.80·37-s − 0.780·41-s + 1.06·43-s + 0.298·45-s + 1.16·47-s + 1/7·49-s + 0.140·51-s + 0.412·53-s + 0.539·55-s + 0.260·59-s − 1.02·61-s + 0.251·63-s + 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 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
−13.24679713226974, −12.78969086835480, −12.28439735610293, −11.92012525795389, −11.33877422347507, −11.02447932345505, −10.48002229350582, −10.22164115178189, −9.521114846981563, −8.992856933819954, −8.492018566138746, −8.120722024957841, −7.433860434393291, −7.196031042581894, −6.337134888839020, −6.153571152401997, −5.397172576588621, −5.175772382741088, −4.394212809555117, −4.107613572011825, −3.125886160788058, −2.767402555844094, −2.420357882368073, −1.282158740507632, −0.6372891695764402, 0,
0.6372891695764402, 1.282158740507632, 2.420357882368073, 2.767402555844094, 3.125886160788058, 4.107613572011825, 4.394212809555117, 5.175772382741088, 5.397172576588621, 6.153571152401997, 6.337134888839020, 7.196031042581894, 7.433860434393291, 8.120722024957841, 8.492018566138746, 8.992856933819954, 9.521114846981563, 10.22164115178189, 10.48002229350582, 11.02447932345505, 11.33877422347507, 11.92012525795389, 12.28439735610293, 12.78969086835480, 13.24679713226974
