| L(s) = 1 | + 3-s − 0.561·5-s + 9-s + 2.56·11-s + 4.56·13-s − 0.561·15-s + 17-s − 7.68·19-s + 6.56·23-s − 4.68·25-s + 27-s + 8.24·29-s + 5.12·31-s + 2.56·33-s + 3.12·37-s + 4.56·39-s + 0.561·41-s + 7.68·43-s − 0.561·45-s + 2.87·47-s − 7·49-s + 51-s − 4.24·53-s − 1.43·55-s − 7.68·57-s + 1.12·59-s + 0.876·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.251·5-s + 0.333·9-s + 0.772·11-s + 1.26·13-s − 0.144·15-s + 0.242·17-s − 1.76·19-s + 1.36·23-s − 0.936·25-s + 0.192·27-s + 1.53·29-s + 0.920·31-s + 0.445·33-s + 0.513·37-s + 0.730·39-s + 0.0876·41-s + 1.17·43-s − 0.0837·45-s + 0.419·47-s − 49-s + 0.140·51-s − 0.583·53-s − 0.193·55-s − 1.01·57-s + 0.146·59-s + 0.112·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.978766995\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.978766995\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 0.561T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 2.56T + 11T^{2} \) |
| 13 | \( 1 - 4.56T + 13T^{2} \) |
| 19 | \( 1 + 7.68T + 19T^{2} \) |
| 23 | \( 1 - 6.56T + 23T^{2} \) |
| 29 | \( 1 - 8.24T + 29T^{2} \) |
| 31 | \( 1 - 5.12T + 31T^{2} \) |
| 37 | \( 1 - 3.12T + 37T^{2} \) |
| 41 | \( 1 - 0.561T + 41T^{2} \) |
| 43 | \( 1 - 7.68T + 43T^{2} \) |
| 47 | \( 1 - 2.87T + 47T^{2} \) |
| 53 | \( 1 + 4.24T + 53T^{2} \) |
| 59 | \( 1 - 1.12T + 59T^{2} \) |
| 61 | \( 1 - 0.876T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 - 4.24T + 73T^{2} \) |
| 79 | \( 1 + 15.3T + 79T^{2} \) |
| 83 | \( 1 - 9.12T + 83T^{2} \) |
| 89 | \( 1 - 7.12T + 89T^{2} \) |
| 97 | \( 1 + 11.1T + 97T^{2} \) |
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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
−10.27130891403138175110194025168, −9.182095494256083697936527309150, −8.592584728026646421335721379551, −7.87474422956294230035221221222, −6.68672170180493886756721043112, −6.09897170988790082049841781830, −4.57825805922526271821601755271, −3.84445324458262372292878531070, −2.71643293968403672919408683305, −1.25684556579012388008287058863,
1.25684556579012388008287058863, 2.71643293968403672919408683305, 3.84445324458262372292878531070, 4.57825805922526271821601755271, 6.09897170988790082049841781830, 6.68672170180493886756721043112, 7.87474422956294230035221221222, 8.592584728026646421335721379551, 9.182095494256083697936527309150, 10.27130891403138175110194025168
