| L(s) = 1 | + 64·2-s + 4.09e3·4-s + 3.67e4·5-s + 1.17e5·7-s + 2.62e5·8-s + 2.35e6·10-s − 1.03e7·11-s + 1.95e7·13-s + 7.52e6·14-s + 1.67e7·16-s − 1.89e8·17-s + 7.97e7·19-s + 1.50e8·20-s − 6.64e8·22-s − 2.00e8·23-s + 1.27e8·25-s + 1.25e9·26-s + 4.81e8·28-s − 4.75e9·29-s − 9.03e9·31-s + 1.07e9·32-s − 1.21e10·34-s + 4.32e9·35-s + 4.55e8·37-s + 5.10e9·38-s + 9.62e9·40-s − 4.00e10·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.05·5-s + 0.377·7-s + 0.353·8-s + 0.743·10-s − 1.76·11-s + 1.12·13-s + 0.267·14-s + 0.250·16-s − 1.90·17-s + 0.388·19-s + 0.525·20-s − 1.25·22-s − 0.283·23-s + 0.104·25-s + 0.795·26-s + 0.188·28-s − 1.48·29-s − 1.82·31-s + 0.176·32-s − 1.34·34-s + 0.397·35-s + 0.0291·37-s + 0.274·38-s + 0.371·40-s − 1.31·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 64T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 1.17e5T \) |
| good | 5 | \( 1 - 3.67e4T + 1.22e9T^{2} \) |
| 11 | \( 1 + 1.03e7T + 3.45e13T^{2} \) |
| 13 | \( 1 - 1.95e7T + 3.02e14T^{2} \) |
| 17 | \( 1 + 1.89e8T + 9.90e15T^{2} \) |
| 19 | \( 1 - 7.97e7T + 4.20e16T^{2} \) |
| 23 | \( 1 + 2.00e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + 4.75e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + 9.03e9T + 2.44e19T^{2} \) |
| 37 | \( 1 - 4.55e8T + 2.43e20T^{2} \) |
| 41 | \( 1 + 4.00e10T + 9.25e20T^{2} \) |
| 43 | \( 1 - 1.11e10T + 1.71e21T^{2} \) |
| 47 | \( 1 - 7.85e10T + 5.46e21T^{2} \) |
| 53 | \( 1 + 6.64e10T + 2.60e22T^{2} \) |
| 59 | \( 1 + 1.17e11T + 1.04e23T^{2} \) |
| 61 | \( 1 - 4.37e10T + 1.61e23T^{2} \) |
| 67 | \( 1 - 1.41e12T + 5.48e23T^{2} \) |
| 71 | \( 1 - 1.68e12T + 1.16e24T^{2} \) |
| 73 | \( 1 - 1.17e12T + 1.67e24T^{2} \) |
| 79 | \( 1 + 4.14e12T + 4.66e24T^{2} \) |
| 83 | \( 1 + 3.82e11T + 8.87e24T^{2} \) |
| 89 | \( 1 + 6.28e11T + 2.19e25T^{2} \) |
| 97 | \( 1 + 1.20e13T + 6.73e25T^{2} \) |
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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
−10.72480165137064652480192392491, −9.424516681885508021246457819420, −8.242967662686097756474653937339, −7.00679813413355615074025951376, −5.78659637278635721666249107397, −5.18543999935526513943514895050, −3.82660216826688716401964814543, −2.41516111160648372263830332141, −1.74498912130056286952477333493, 0,
1.74498912130056286952477333493, 2.41516111160648372263830332141, 3.82660216826688716401964814543, 5.18543999935526513943514895050, 5.78659637278635721666249107397, 7.00679813413355615074025951376, 8.242967662686097756474653937339, 9.424516681885508021246457819420, 10.72480165137064652480192392491
