| L(s) = 1 | + 28.4·5-s − 49·7-s + 424.·11-s + 508.·13-s + 539.·17-s − 2.60e3·19-s + 261.·23-s − 2.31e3·25-s − 6.87e3·29-s − 5.68e3·31-s − 1.39e3·35-s + 4.90e3·37-s + 5.72e3·41-s + 1.73e3·43-s − 1.01e4·47-s + 2.40e3·49-s + 3.11e4·53-s + 1.20e4·55-s − 3.88e4·59-s + 1.36e4·61-s + 1.44e4·65-s + 3.07e4·67-s − 4.56e4·71-s + 2.17e4·73-s − 2.07e4·77-s − 3.22e4·79-s − 4.66e4·83-s + ⋯ |
| L(s) = 1 | + 0.508·5-s − 0.377·7-s + 1.05·11-s + 0.834·13-s + 0.453·17-s − 1.65·19-s + 0.102·23-s − 0.741·25-s − 1.51·29-s − 1.06·31-s − 0.192·35-s + 0.589·37-s + 0.531·41-s + 0.143·43-s − 0.670·47-s + 0.142·49-s + 1.52·53-s + 0.537·55-s − 1.45·59-s + 0.469·61-s + 0.424·65-s + 0.836·67-s − 1.07·71-s + 0.477·73-s − 0.399·77-s − 0.582·79-s − 0.743·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 49T \) |
| good | 5 | \( 1 - 28.4T + 3.12e3T^{2} \) |
| 11 | \( 1 - 424.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 508.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 539.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.60e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 261.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.87e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.68e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.90e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 5.72e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.73e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.01e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.11e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.88e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.36e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.07e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.56e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.17e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.22e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.66e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.37e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.15e5T + 8.58e9T^{2} \) |
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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
−9.019304493183041706347359607186, −8.051104124478749091926629119294, −6.99958280704156735019284454360, −6.17849038766310963069514310135, −5.62445925909879558583495406493, −4.20027496028757755077374614581, −3.58887889145828041099398340385, −2.21441376989172694708514872592, −1.33628438909052813276167543332, 0,
1.33628438909052813276167543332, 2.21441376989172694708514872592, 3.58887889145828041099398340385, 4.20027496028757755077374614581, 5.62445925909879558583495406493, 6.17849038766310963069514310135, 6.99958280704156735019284454360, 8.051104124478749091926629119294, 9.019304493183041706347359607186
