| L(s) = 1 | + 2·3-s + 5-s + 5·7-s + 9-s + 3·11-s − 3·13-s + 2·15-s + 10·21-s − 5·23-s − 4·25-s − 4·27-s + 6·29-s + 6·33-s + 5·35-s − 12·37-s − 6·39-s − 3·41-s + 8·43-s + 45-s − 12·47-s + 18·49-s − 2·53-s + 3·55-s + 9·59-s − 61-s + 5·63-s − 3·65-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s + 1.88·7-s + 1/3·9-s + 0.904·11-s − 0.832·13-s + 0.516·15-s + 2.18·21-s − 1.04·23-s − 4/5·25-s − 0.769·27-s + 1.11·29-s + 1.04·33-s + 0.845·35-s − 1.97·37-s − 0.960·39-s − 0.468·41-s + 1.21·43-s + 0.149·45-s − 1.75·47-s + 18/7·49-s − 0.274·53-s + 0.404·55-s + 1.17·59-s − 0.128·61-s + 0.629·63-s − 0.372·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 976 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 976 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.943884537\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.943884537\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 61 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−9.836514025533174717045754125033, −9.056237507126111645473865568644, −8.259631811168054589484751786913, −7.84997221060806366414385499222, −6.78980578951837536945571914964, −5.53278600499739307280168084058, −4.64912865131610184801613353745, −3.68731906059088689155175333695, −2.30357820078926845828673973788, −1.63976643607280630849823286411,
1.63976643607280630849823286411, 2.30357820078926845828673973788, 3.68731906059088689155175333695, 4.64912865131610184801613353745, 5.53278600499739307280168084058, 6.78980578951837536945571914964, 7.84997221060806366414385499222, 8.259631811168054589484751786913, 9.056237507126111645473865568644, 9.836514025533174717045754125033
