| L(s) = 1 | − 2.05·3-s + 2.67·5-s + 0.688·7-s + 1.20·9-s − 5.36·11-s + 1.75·13-s − 5.48·15-s + 3.58·17-s − 19-s − 1.41·21-s + 1.71·23-s + 2.14·25-s + 3.67·27-s − 10.2·29-s − 1.27·31-s + 11.0·33-s + 1.84·35-s + 1.26·37-s − 3.59·39-s + 9.70·41-s + 2.83·43-s + 3.23·45-s − 7.30·47-s − 6.52·49-s − 7.35·51-s + 53-s − 14.3·55-s + ⋯ |
| L(s) = 1 | − 1.18·3-s + 1.19·5-s + 0.260·7-s + 0.403·9-s − 1.61·11-s + 0.485·13-s − 1.41·15-s + 0.868·17-s − 0.229·19-s − 0.308·21-s + 0.356·23-s + 0.429·25-s + 0.706·27-s − 1.90·29-s − 0.228·31-s + 1.91·33-s + 0.311·35-s + 0.207·37-s − 0.575·39-s + 1.51·41-s + 0.431·43-s + 0.481·45-s − 1.06·47-s − 0.932·49-s − 1.02·51-s + 0.137·53-s − 1.93·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| good | 3 | \( 1 + 2.05T + 3T^{2} \) |
| 5 | \( 1 - 2.67T + 5T^{2} \) |
| 7 | \( 1 - 0.688T + 7T^{2} \) |
| 11 | \( 1 + 5.36T + 11T^{2} \) |
| 13 | \( 1 - 1.75T + 13T^{2} \) |
| 17 | \( 1 - 3.58T + 17T^{2} \) |
| 23 | \( 1 - 1.71T + 23T^{2} \) |
| 29 | \( 1 + 10.2T + 29T^{2} \) |
| 31 | \( 1 + 1.27T + 31T^{2} \) |
| 37 | \( 1 - 1.26T + 37T^{2} \) |
| 41 | \( 1 - 9.70T + 41T^{2} \) |
| 43 | \( 1 - 2.83T + 43T^{2} \) |
| 47 | \( 1 + 7.30T + 47T^{2} \) |
| 59 | \( 1 + 7.26T + 59T^{2} \) |
| 61 | \( 1 + 14.4T + 61T^{2} \) |
| 67 | \( 1 + 4.38T + 67T^{2} \) |
| 71 | \( 1 + 0.131T + 71T^{2} \) |
| 73 | \( 1 - 9.46T + 73T^{2} \) |
| 79 | \( 1 + 7.19T + 79T^{2} \) |
| 83 | \( 1 + 6.22T + 83T^{2} \) |
| 89 | \( 1 - 6.04T + 89T^{2} \) |
| 97 | \( 1 - 10.7T + 97T^{2} \) |
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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
−7.88818729947988447463111148906, −7.38600227034817870540338162296, −6.15931205228343363418281581245, −5.89585541075429373029162548599, −5.27348631131377505741361327037, −4.67247704099795783618592633817, −3.31552242451528130257445798067, −2.33477307279299007602425741658, −1.35011421379858675837205741532, 0,
1.35011421379858675837205741532, 2.33477307279299007602425741658, 3.31552242451528130257445798067, 4.67247704099795783618592633817, 5.27348631131377505741361327037, 5.89585541075429373029162548599, 6.15931205228343363418281581245, 7.38600227034817870540338162296, 7.88818729947988447463111148906
