| L(s) = 1 | − 1.29·3-s + 5-s − 0.227·7-s − 1.32·9-s − 3.34·13-s − 1.29·15-s + 6.04·17-s − 2.22·19-s + 0.294·21-s − 3.47·23-s + 25-s + 5.59·27-s + 0.0307·29-s + 2.21·31-s − 0.227·35-s + 8.74·37-s + 4.33·39-s + 10.2·41-s − 3.74·43-s − 1.32·45-s − 5.31·47-s − 6.94·49-s − 7.83·51-s − 10.2·53-s + 2.87·57-s + 8.34·59-s − 2.24·61-s + ⋯ |
| L(s) = 1 | − 0.747·3-s + 0.447·5-s − 0.0860·7-s − 0.441·9-s − 0.927·13-s − 0.334·15-s + 1.46·17-s − 0.509·19-s + 0.0643·21-s − 0.725·23-s + 0.200·25-s + 1.07·27-s + 0.00571·29-s + 0.398·31-s − 0.0385·35-s + 1.43·37-s + 0.693·39-s + 1.59·41-s − 0.570·43-s − 0.197·45-s − 0.775·47-s − 0.992·49-s − 1.09·51-s − 1.40·53-s + 0.381·57-s + 1.08·59-s − 0.287·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 1.29T + 3T^{2} \) |
| 7 | \( 1 + 0.227T + 7T^{2} \) |
| 13 | \( 1 + 3.34T + 13T^{2} \) |
| 17 | \( 1 - 6.04T + 17T^{2} \) |
| 19 | \( 1 + 2.22T + 19T^{2} \) |
| 23 | \( 1 + 3.47T + 23T^{2} \) |
| 29 | \( 1 - 0.0307T + 29T^{2} \) |
| 31 | \( 1 - 2.21T + 31T^{2} \) |
| 37 | \( 1 - 8.74T + 37T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 + 3.74T + 43T^{2} \) |
| 47 | \( 1 + 5.31T + 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 - 8.34T + 59T^{2} \) |
| 61 | \( 1 + 2.24T + 61T^{2} \) |
| 67 | \( 1 - 3.47T + 67T^{2} \) |
| 71 | \( 1 + 15.0T + 71T^{2} \) |
| 73 | \( 1 + 3.66T + 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 - 1.52T + 83T^{2} \) |
| 89 | \( 1 + 10.0T + 89T^{2} \) |
| 97 | \( 1 + 8.33T + 97T^{2} \) |
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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
−7.896233924131615772102594211675, −7.16188350443825337913299711824, −6.14930378360452239016544511183, −5.90986479284081894543370576759, −5.04404883892028181652436424424, −4.38738630530687487003304686354, −3.19435941443248751708236777271, −2.45479722197784585808057991420, −1.24016304496842748810740391226, 0,
1.24016304496842748810740391226, 2.45479722197784585808057991420, 3.19435941443248751708236777271, 4.38738630530687487003304686354, 5.04404883892028181652436424424, 5.90986479284081894543370576759, 6.14930378360452239016544511183, 7.16188350443825337913299711824, 7.896233924131615772102594211675
