| L(s) = 1 | − 0.870·5-s − 5.07·11-s − 7.17·17-s + 1.58·19-s − 3.84·23-s − 4.24·25-s + 2.46·29-s + 3·31-s + 5.24·37-s − 4.56·41-s + 0.242·43-s + 5.43·47-s + 10.1·53-s + 4.41·55-s + 10.6·59-s + 11.6·61-s − 8.48·67-s − 2.61·71-s + 7.41·73-s + 0.242·79-s − 1.74·83-s + 6.24·85-s + 4.56·89-s − 1.38·95-s − 5.65·97-s + 3.69·101-s + 12.1·103-s + ⋯ |
| L(s) = 1 | − 0.389·5-s − 1.52·11-s − 1.73·17-s + 0.363·19-s − 0.801·23-s − 0.848·25-s + 0.457·29-s + 0.538·31-s + 0.861·37-s − 0.712·41-s + 0.0370·43-s + 0.792·47-s + 1.39·53-s + 0.595·55-s + 1.38·59-s + 1.49·61-s − 1.03·67-s − 0.309·71-s + 0.867·73-s + 0.0272·79-s − 0.191·83-s + 0.677·85-s + 0.483·89-s − 0.141·95-s − 0.574·97-s + 0.367·101-s + 1.19·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.050485509\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.050485509\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 0.870T + 5T^{2} \) |
| 11 | \( 1 + 5.07T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 7.17T + 17T^{2} \) |
| 19 | \( 1 - 1.58T + 19T^{2} \) |
| 23 | \( 1 + 3.84T + 23T^{2} \) |
| 29 | \( 1 - 2.46T + 29T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 - 5.24T + 37T^{2} \) |
| 41 | \( 1 + 4.56T + 41T^{2} \) |
| 43 | \( 1 - 0.242T + 43T^{2} \) |
| 47 | \( 1 - 5.43T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 - 10.6T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 + 8.48T + 67T^{2} \) |
| 71 | \( 1 + 2.61T + 71T^{2} \) |
| 73 | \( 1 - 7.41T + 73T^{2} \) |
| 79 | \( 1 - 0.242T + 79T^{2} \) |
| 83 | \( 1 + 1.74T + 83T^{2} \) |
| 89 | \( 1 - 4.56T + 89T^{2} \) |
| 97 | \( 1 + 5.65T + 97T^{2} \) |
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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
−8.229462740824048334330809242789, −7.52140023587285103588415324945, −6.85543359743328888154658280883, −5.99829593932807403562182674549, −5.27334747557283777651625338078, −4.48423555248463734100356260311, −3.81234310588869062998767858539, −2.66859372898711396191363325630, −2.12528456425412554652149259691, −0.52296799950069717061562912262,
0.52296799950069717061562912262, 2.12528456425412554652149259691, 2.66859372898711396191363325630, 3.81234310588869062998767858539, 4.48423555248463734100356260311, 5.27334747557283777651625338078, 5.99829593932807403562182674549, 6.85543359743328888154658280883, 7.52140023587285103588415324945, 8.229462740824048334330809242789
