| L(s) = 1 | − 3-s + 1.56·5-s − 3.21·7-s + 9-s + 5.96·11-s + 2.81·13-s − 1.56·15-s + 4.33·17-s − 4.82·19-s + 3.21·21-s − 3.36·23-s − 2.53·25-s − 27-s − 8.76·29-s − 2.34·31-s − 5.96·33-s − 5.04·35-s + 7.28·37-s − 2.81·39-s − 1.39·41-s − 2.86·43-s + 1.56·45-s − 9.87·47-s + 3.33·49-s − 4.33·51-s − 11.2·53-s + 9.36·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.701·5-s − 1.21·7-s + 0.333·9-s + 1.79·11-s + 0.780·13-s − 0.405·15-s + 1.05·17-s − 1.10·19-s + 0.701·21-s − 0.701·23-s − 0.507·25-s − 0.192·27-s − 1.62·29-s − 0.420·31-s − 1.03·33-s − 0.852·35-s + 1.19·37-s − 0.450·39-s − 0.217·41-s − 0.436·43-s + 0.233·45-s − 1.44·47-s + 0.476·49-s − 0.606·51-s − 1.54·53-s + 1.26·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 - 1.56T + 5T^{2} \) |
| 7 | \( 1 + 3.21T + 7T^{2} \) |
| 11 | \( 1 - 5.96T + 11T^{2} \) |
| 13 | \( 1 - 2.81T + 13T^{2} \) |
| 17 | \( 1 - 4.33T + 17T^{2} \) |
| 19 | \( 1 + 4.82T + 19T^{2} \) |
| 23 | \( 1 + 3.36T + 23T^{2} \) |
| 29 | \( 1 + 8.76T + 29T^{2} \) |
| 31 | \( 1 + 2.34T + 31T^{2} \) |
| 37 | \( 1 - 7.28T + 37T^{2} \) |
| 41 | \( 1 + 1.39T + 41T^{2} \) |
| 43 | \( 1 + 2.86T + 43T^{2} \) |
| 47 | \( 1 + 9.87T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 - 1.96T + 59T^{2} \) |
| 61 | \( 1 + 3.76T + 61T^{2} \) |
| 67 | \( 1 + 11.6T + 67T^{2} \) |
| 71 | \( 1 - 4.01T + 71T^{2} \) |
| 73 | \( 1 - 8.49T + 73T^{2} \) |
| 79 | \( 1 + 3.69T + 79T^{2} \) |
| 83 | \( 1 + 8.33T + 83T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 - 0.582T + 97T^{2} \) |
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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
−7.31269070696769453430352253416, −6.44079497608331485641829092996, −6.16984428862573189874656332027, −5.78667619651104799517094899651, −4.63200692095439271552272437405, −3.73952299585523074041028566381, −3.40427519583896698997518115994, −1.99013232784379187151706134728, −1.31408101516674436445354031123, 0,
1.31408101516674436445354031123, 1.99013232784379187151706134728, 3.40427519583896698997518115994, 3.73952299585523074041028566381, 4.63200692095439271552272437405, 5.78667619651104799517094899651, 6.16984428862573189874656332027, 6.44079497608331485641829092996, 7.31269070696769453430352253416
