| L(s) = 1 | + 1.52·3-s + 2.88·5-s − 0.672·9-s − 11-s − 1.40·13-s + 4.40·15-s + 2.97·17-s + 5.25·19-s + 2.65·23-s + 3.33·25-s − 5.60·27-s + 3.44·29-s + 2.46·31-s − 1.52·33-s − 1.88·37-s − 2.14·39-s − 5.68·41-s + 10.6·43-s − 1.94·45-s + 8.02·47-s + 4.53·51-s + 9.94·53-s − 2.88·55-s + 8.02·57-s − 1.17·59-s − 8.46·61-s − 4.06·65-s + ⋯ |
| L(s) = 1 | + 0.880·3-s + 1.29·5-s − 0.224·9-s − 0.301·11-s − 0.390·13-s + 1.13·15-s + 0.721·17-s + 1.20·19-s + 0.554·23-s + 0.666·25-s − 1.07·27-s + 0.639·29-s + 0.442·31-s − 0.265·33-s − 0.309·37-s − 0.343·39-s − 0.887·41-s + 1.62·43-s − 0.289·45-s + 1.17·47-s + 0.635·51-s + 1.36·53-s − 0.389·55-s + 1.06·57-s − 0.153·59-s − 1.08·61-s − 0.504·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.834521808\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.834521808\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 1.52T + 3T^{2} \) |
| 5 | \( 1 - 2.88T + 5T^{2} \) |
| 13 | \( 1 + 1.40T + 13T^{2} \) |
| 17 | \( 1 - 2.97T + 17T^{2} \) |
| 19 | \( 1 - 5.25T + 19T^{2} \) |
| 23 | \( 1 - 2.65T + 23T^{2} \) |
| 29 | \( 1 - 3.44T + 29T^{2} \) |
| 31 | \( 1 - 2.46T + 31T^{2} \) |
| 37 | \( 1 + 1.88T + 37T^{2} \) |
| 41 | \( 1 + 5.68T + 41T^{2} \) |
| 43 | \( 1 - 10.6T + 43T^{2} \) |
| 47 | \( 1 - 8.02T + 47T^{2} \) |
| 53 | \( 1 - 9.94T + 53T^{2} \) |
| 59 | \( 1 + 1.17T + 59T^{2} \) |
| 61 | \( 1 + 8.46T + 61T^{2} \) |
| 67 | \( 1 - 1.02T + 67T^{2} \) |
| 71 | \( 1 + 11.9T + 71T^{2} \) |
| 73 | \( 1 - 5.90T + 73T^{2} \) |
| 79 | \( 1 - 8.60T + 79T^{2} \) |
| 83 | \( 1 - 6.32T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 + 0.175T + 97T^{2} \) |
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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
−7.75027983090126534242993645153, −7.25158360440711590054053031237, −6.34345249924960621869995473663, −5.55384432979350552351888650406, −5.26866799008571399885286514145, −4.18177860544914231556576922463, −3.13767143285585113845011118073, −2.72050721548980966694943102722, −1.93403646890101884161030237780, −0.936071559048242997613623229886,
0.936071559048242997613623229886, 1.93403646890101884161030237780, 2.72050721548980966694943102722, 3.13767143285585113845011118073, 4.18177860544914231556576922463, 5.26866799008571399885286514145, 5.55384432979350552351888650406, 6.34345249924960621869995473663, 7.25158360440711590054053031237, 7.75027983090126534242993645153
