| L(s) = 1 | − 2.41·3-s + 3.41·5-s − 3.41·7-s + 2.82·9-s + 4.24·11-s − 1.82·13-s − 8.24·15-s − 5.41·17-s − 4.82·19-s + 8.24·21-s − 23-s + 6.65·25-s + 0.414·27-s + 5·29-s − 0.757·31-s − 10.2·33-s − 11.6·35-s + 9.07·37-s + 4.41·39-s − 12.6·41-s − 4·43-s + 9.65·45-s + 3.24·47-s + 4.65·49-s + 13.0·51-s − 13.3·53-s + 14.4·55-s + ⋯ |
| L(s) = 1 | − 1.39·3-s + 1.52·5-s − 1.29·7-s + 0.942·9-s + 1.27·11-s − 0.507·13-s − 2.12·15-s − 1.31·17-s − 1.10·19-s + 1.79·21-s − 0.208·23-s + 1.33·25-s + 0.0797·27-s + 0.928·29-s − 0.136·31-s − 1.78·33-s − 1.97·35-s + 1.49·37-s + 0.706·39-s − 1.97·41-s − 0.609·43-s + 1.43·45-s + 0.472·47-s + 0.665·49-s + 1.83·51-s − 1.82·53-s + 1.95·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 2.41T + 3T^{2} \) |
| 5 | \( 1 - 3.41T + 5T^{2} \) |
| 7 | \( 1 + 3.41T + 7T^{2} \) |
| 11 | \( 1 - 4.24T + 11T^{2} \) |
| 13 | \( 1 + 1.82T + 13T^{2} \) |
| 17 | \( 1 + 5.41T + 17T^{2} \) |
| 19 | \( 1 + 4.82T + 19T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 + 0.757T + 31T^{2} \) |
| 37 | \( 1 - 9.07T + 37T^{2} \) |
| 41 | \( 1 + 12.6T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 3.24T + 47T^{2} \) |
| 53 | \( 1 + 13.3T + 53T^{2} \) |
| 59 | \( 1 + 6.82T + 59T^{2} \) |
| 61 | \( 1 + 7.65T + 61T^{2} \) |
| 67 | \( 1 - 5.89T + 67T^{2} \) |
| 71 | \( 1 - 6.41T + 71T^{2} \) |
| 73 | \( 1 - 2.17T + 73T^{2} \) |
| 79 | \( 1 - 2.48T + 79T^{2} \) |
| 83 | \( 1 + 14.7T + 83T^{2} \) |
| 89 | \( 1 + 8T + 89T^{2} \) |
| 97 | \( 1 + 9.89T + 97T^{2} \) |
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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
−9.456159427590212657315575217927, −8.567465982923886792483285895032, −6.79934804240066478870929483094, −6.43662045629618641279198563567, −6.15217878295515502687569788355, −5.08648939719730197012115311983, −4.22843299771050632436970299366, −2.76417225677206653802405433818, −1.57599364917149848294130471831, 0,
1.57599364917149848294130471831, 2.76417225677206653802405433818, 4.22843299771050632436970299366, 5.08648939719730197012115311983, 6.15217878295515502687569788355, 6.43662045629618641279198563567, 6.79934804240066478870929483094, 8.567465982923886792483285895032, 9.456159427590212657315575217927
