| L(s) = 1 | − 3-s − 5-s + 2·7-s + 9-s + 6·11-s − 13-s + 15-s + 17-s − 6·19-s − 2·21-s − 6·23-s + 25-s − 27-s + 5·29-s − 5·31-s − 6·33-s − 2·35-s + 8·37-s + 39-s − 6·41-s − 10·43-s − 45-s − 3·49-s − 51-s − 3·53-s − 6·55-s + 6·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s + 1.80·11-s − 0.277·13-s + 0.258·15-s + 0.242·17-s − 1.37·19-s − 0.436·21-s − 1.25·23-s + 1/5·25-s − 0.192·27-s + 0.928·29-s − 0.898·31-s − 1.04·33-s − 0.338·35-s + 1.31·37-s + 0.160·39-s − 0.937·41-s − 1.52·43-s − 0.149·45-s − 3/7·49-s − 0.140·51-s − 0.412·53-s − 0.809·55-s + 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.113271895\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.113271895\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 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
−12.76347774159429, −12.48767609863109, −11.92632544617903, −11.60932214456044, −11.35659390448493, −10.72143001815138, −10.26423584021387, −9.778746888620136, −9.256405275565979, −8.619155036465435, −8.381552420980365, −7.740121357186988, −7.296243843874311, −6.632558923014470, −6.278652956570568, −5.957272276650311, −5.113563973302238, −4.529767809364588, −4.363259440760985, −3.712414675325315, −3.205775844456273, −2.252237580190567, −1.634645477227086, −1.294002598041540, −0.3115903026975533,
0.3115903026975533, 1.294002598041540, 1.634645477227086, 2.252237580190567, 3.205775844456273, 3.712414675325315, 4.363259440760985, 4.529767809364588, 5.113563973302238, 5.957272276650311, 6.278652956570568, 6.632558923014470, 7.296243843874311, 7.740121357186988, 8.381552420980365, 8.619155036465435, 9.256405275565979, 9.778746888620136, 10.26423584021387, 10.72143001815138, 11.35659390448493, 11.60932214456044, 11.92632544617903, 12.48767609863109, 12.76347774159429
