| L(s) = 1 | − 3-s − 0.561·5-s + 7-s + 9-s − 2.56·11-s + 4.56·13-s + 0.561·15-s + 17-s + 0.561·19-s − 21-s − 2.56·23-s − 4.68·25-s − 27-s − 1.12·29-s + 5.12·31-s + 2.56·33-s − 0.561·35-s + 7.12·37-s − 4.56·39-s + 4.56·41-s − 1.43·43-s − 0.561·45-s − 5.12·47-s + 49-s − 51-s + 6·53-s + 1.43·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.251·5-s + 0.377·7-s + 0.333·9-s − 0.772·11-s + 1.26·13-s + 0.144·15-s + 0.242·17-s + 0.128·19-s − 0.218·21-s − 0.534·23-s − 0.936·25-s − 0.192·27-s − 0.208·29-s + 0.920·31-s + 0.445·33-s − 0.0949·35-s + 1.17·37-s − 0.730·39-s + 0.712·41-s − 0.219·43-s − 0.0837·45-s − 0.747·47-s + 0.142·49-s − 0.140·51-s + 0.824·53-s + 0.193·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.485937780\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.485937780\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 0.561T + 5T^{2} \) |
| 11 | \( 1 + 2.56T + 11T^{2} \) |
| 13 | \( 1 - 4.56T + 13T^{2} \) |
| 19 | \( 1 - 0.561T + 19T^{2} \) |
| 23 | \( 1 + 2.56T + 23T^{2} \) |
| 29 | \( 1 + 1.12T + 29T^{2} \) |
| 31 | \( 1 - 5.12T + 31T^{2} \) |
| 37 | \( 1 - 7.12T + 37T^{2} \) |
| 41 | \( 1 - 4.56T + 41T^{2} \) |
| 43 | \( 1 + 1.43T + 43T^{2} \) |
| 47 | \( 1 + 5.12T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 - 14.2T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 + 5.12T + 73T^{2} \) |
| 79 | \( 1 + 7.36T + 79T^{2} \) |
| 83 | \( 1 - 5.12T + 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 + 14.2T + 97T^{2} \) |
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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
−8.017119626327761558067972188882, −7.57261570996245481318485659009, −6.56916198784719116514440158298, −5.93211149174073308880979943336, −5.36413881549599092246633704766, −4.44045270636198190629524015318, −3.83252769944893195986423059417, −2.81621326117819935927835042674, −1.73392163390563410545671193219, −0.68245589433683787157343506089,
0.68245589433683787157343506089, 1.73392163390563410545671193219, 2.81621326117819935927835042674, 3.83252769944893195986423059417, 4.44045270636198190629524015318, 5.36413881549599092246633704766, 5.93211149174073308880979943336, 6.56916198784719116514440158298, 7.57261570996245481318485659009, 8.017119626327761558067972188882
