| L(s) = 1 | + 3.30·3-s − 2.30·5-s − 2.30·7-s + 7.90·9-s + 5.60·11-s + 0.697·13-s − 7.60·15-s − 1.30·19-s − 7.60·21-s + 4.30·23-s + 0.302·25-s + 16.2·27-s − 0.697·29-s − 4.21·31-s + 18.5·33-s + 5.30·35-s + 8.60·37-s + 2.30·39-s + 6·41-s − 4.21·43-s − 18.2·45-s + 11.6·47-s − 1.69·49-s + 3.30·53-s − 12.9·55-s − 4.30·57-s + 3.21·59-s + ⋯ |
| L(s) = 1 | + 1.90·3-s − 1.02·5-s − 0.870·7-s + 2.63·9-s + 1.69·11-s + 0.193·13-s − 1.96·15-s − 0.298·19-s − 1.65·21-s + 0.897·23-s + 0.0605·25-s + 3.11·27-s − 0.129·29-s − 0.756·31-s + 3.22·33-s + 0.896·35-s + 1.41·37-s + 0.368·39-s + 0.937·41-s − 0.642·43-s − 2.71·45-s + 1.69·47-s − 0.242·49-s + 0.453·53-s − 1.74·55-s − 0.569·57-s + 0.418·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.173369567\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.173369567\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - 3.30T + 3T^{2} \) |
| 5 | \( 1 + 2.30T + 5T^{2} \) |
| 7 | \( 1 + 2.30T + 7T^{2} \) |
| 11 | \( 1 - 5.60T + 11T^{2} \) |
| 13 | \( 1 - 0.697T + 13T^{2} \) |
| 19 | \( 1 + 1.30T + 19T^{2} \) |
| 23 | \( 1 - 4.30T + 23T^{2} \) |
| 29 | \( 1 + 0.697T + 29T^{2} \) |
| 31 | \( 1 + 4.21T + 31T^{2} \) |
| 37 | \( 1 - 8.60T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 4.21T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 - 3.30T + 53T^{2} \) |
| 59 | \( 1 - 3.21T + 59T^{2} \) |
| 61 | \( 1 + 1.60T + 61T^{2} \) |
| 67 | \( 1 - 0.605T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 - 2.39T + 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 + 8.69T + 83T^{2} \) |
| 89 | \( 1 - 7.21T + 89T^{2} \) |
| 97 | \( 1 - 16.1T + 97T^{2} \) |
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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
−9.169165485870980792374468580947, −8.332286340371683642725627362042, −7.51799185001099265828331112898, −7.01847261601045011402294796266, −6.17602556538533355834081368692, −4.46020974888267058317785843314, −3.84900269151386815076817170358, −3.39280934842251003212584781937, −2.41834151267586668373319769326, −1.13902694643739783132310572933,
1.13902694643739783132310572933, 2.41834151267586668373319769326, 3.39280934842251003212584781937, 3.84900269151386815076817170358, 4.46020974888267058317785843314, 6.17602556538533355834081368692, 7.01847261601045011402294796266, 7.51799185001099265828331112898, 8.332286340371683642725627362042, 9.169165485870980792374468580947
