| L(s) = 1 | + 0.219·3-s − 2.01·5-s − 0.320·7-s − 2.95·9-s + 2.72·11-s − 1.65·13-s − 0.441·15-s + 1.51·17-s + 5.02·19-s − 0.0701·21-s − 4.75·23-s − 0.939·25-s − 1.30·27-s + 6.26·29-s − 2.15·31-s + 0.596·33-s + 0.645·35-s + 10.3·37-s − 0.362·39-s − 3.28·43-s + 5.94·45-s + 1.14·47-s − 6.89·49-s + 0.332·51-s − 3.25·53-s − 5.48·55-s + 1.10·57-s + ⋯ |
| L(s) = 1 | + 0.126·3-s − 0.901·5-s − 0.121·7-s − 0.983·9-s + 0.820·11-s − 0.458·13-s − 0.113·15-s + 0.368·17-s + 1.15·19-s − 0.0153·21-s − 0.990·23-s − 0.187·25-s − 0.250·27-s + 1.16·29-s − 0.386·31-s + 0.103·33-s + 0.109·35-s + 1.69·37-s − 0.0580·39-s − 0.500·43-s + 0.886·45-s + 0.166·47-s − 0.985·49-s + 0.0465·51-s − 0.446·53-s − 0.739·55-s + 0.145·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6724 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 41 | \( 1 \) |
| good | 3 | \( 1 - 0.219T + 3T^{2} \) |
| 5 | \( 1 + 2.01T + 5T^{2} \) |
| 7 | \( 1 + 0.320T + 7T^{2} \) |
| 11 | \( 1 - 2.72T + 11T^{2} \) |
| 13 | \( 1 + 1.65T + 13T^{2} \) |
| 17 | \( 1 - 1.51T + 17T^{2} \) |
| 19 | \( 1 - 5.02T + 19T^{2} \) |
| 23 | \( 1 + 4.75T + 23T^{2} \) |
| 29 | \( 1 - 6.26T + 29T^{2} \) |
| 31 | \( 1 + 2.15T + 31T^{2} \) |
| 37 | \( 1 - 10.3T + 37T^{2} \) |
| 43 | \( 1 + 3.28T + 43T^{2} \) |
| 47 | \( 1 - 1.14T + 47T^{2} \) |
| 53 | \( 1 + 3.25T + 53T^{2} \) |
| 59 | \( 1 - 12.4T + 59T^{2} \) |
| 61 | \( 1 + 3.52T + 61T^{2} \) |
| 67 | \( 1 - 3.14T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 + 6.77T + 73T^{2} \) |
| 79 | \( 1 - 9.88T + 79T^{2} \) |
| 83 | \( 1 - 3.77T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 + 5.32T + 97T^{2} \) |
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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
−7.83828837260113688889754227510, −6.99785188406114630111547151535, −6.22093129096591823612755985427, −5.55858108868515565067106799880, −4.67835854409541256301827782390, −3.90028612974063754493330184937, −3.24722513386213768519231645114, −2.46110018577504780563215869174, −1.16529772970521914694723772756, 0,
1.16529772970521914694723772756, 2.46110018577504780563215869174, 3.24722513386213768519231645114, 3.90028612974063754493330184937, 4.67835854409541256301827782390, 5.55858108868515565067106799880, 6.22093129096591823612755985427, 6.99785188406114630111547151535, 7.83828837260113688889754227510
