| L(s) = 1 | + 1.39·5-s + 2.18·7-s − 11-s − 13-s − 1.72·17-s − 6.58·19-s + 6.78·23-s − 3.04·25-s − 0.646·29-s − 10.3·31-s + 3.05·35-s − 2.60·37-s − 7.90·41-s − 11.5·43-s − 4.79·47-s − 2.22·49-s + 1.78·53-s − 1.39·55-s + 13.6·59-s + 1.87·61-s − 1.39·65-s − 9.91·67-s − 8.17·71-s + 11.3·73-s − 2.18·77-s + 14.0·79-s − 2.46·83-s + ⋯ |
| L(s) = 1 | + 0.625·5-s + 0.825·7-s − 0.301·11-s − 0.277·13-s − 0.417·17-s − 1.51·19-s + 1.41·23-s − 0.609·25-s − 0.120·29-s − 1.85·31-s + 0.516·35-s − 0.427·37-s − 1.23·41-s − 1.76·43-s − 0.699·47-s − 0.318·49-s + 0.245·53-s − 0.188·55-s + 1.77·59-s + 0.239·61-s − 0.173·65-s − 1.21·67-s − 0.970·71-s + 1.32·73-s − 0.248·77-s + 1.58·79-s − 0.270·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 1.39T + 5T^{2} \) |
| 7 | \( 1 - 2.18T + 7T^{2} \) |
| 17 | \( 1 + 1.72T + 17T^{2} \) |
| 19 | \( 1 + 6.58T + 19T^{2} \) |
| 23 | \( 1 - 6.78T + 23T^{2} \) |
| 29 | \( 1 + 0.646T + 29T^{2} \) |
| 31 | \( 1 + 10.3T + 31T^{2} \) |
| 37 | \( 1 + 2.60T + 37T^{2} \) |
| 41 | \( 1 + 7.90T + 41T^{2} \) |
| 43 | \( 1 + 11.5T + 43T^{2} \) |
| 47 | \( 1 + 4.79T + 47T^{2} \) |
| 53 | \( 1 - 1.78T + 53T^{2} \) |
| 59 | \( 1 - 13.6T + 59T^{2} \) |
| 61 | \( 1 - 1.87T + 61T^{2} \) |
| 67 | \( 1 + 9.91T + 67T^{2} \) |
| 71 | \( 1 + 8.17T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 - 14.0T + 79T^{2} \) |
| 83 | \( 1 + 2.46T + 83T^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 + 1.39T + 97T^{2} \) |
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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
−7.977115419487527183263518577524, −7.00094499655829553875130930027, −6.57134559768075669428713399474, −5.45883816246284311749369833521, −5.10085159697058302390424779468, −4.21973490087248042157095068150, −3.26388395625333571903631386765, −2.13731414737750632900375740191, −1.64299688636468021405116856550, 0,
1.64299688636468021405116856550, 2.13731414737750632900375740191, 3.26388395625333571903631386765, 4.21973490087248042157095068150, 5.10085159697058302390424779468, 5.45883816246284311749369833521, 6.57134559768075669428713399474, 7.00094499655829553875130930027, 7.977115419487527183263518577524
