| L(s) = 1 | − 64·2-s − 1.59e3·3-s + 4.09e3·4-s − 5.71e4·5-s + 1.02e5·6-s + 5.50e5·7-s − 2.62e5·8-s + 9.53e5·9-s + 3.65e6·10-s − 1.77e6·11-s − 6.53e6·12-s + 2.35e7·13-s − 3.52e7·14-s + 9.12e7·15-s + 1.67e7·16-s + 1.01e8·17-s − 6.10e7·18-s − 1.12e8·19-s − 2.34e8·20-s − 8.78e8·21-s + 1.13e8·22-s − 6.98e8·23-s + 4.18e8·24-s + 2.04e9·25-s − 1.50e9·26-s + 1.02e9·27-s + 2.25e9·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.26·3-s + 0.5·4-s − 1.63·5-s + 0.893·6-s + 1.76·7-s − 0.353·8-s + 0.597·9-s + 1.15·10-s − 0.301·11-s − 0.632·12-s + 1.35·13-s − 1.24·14-s + 2.06·15-s + 0.250·16-s + 1.02·17-s − 0.422·18-s − 0.549·19-s − 0.818·20-s − 2.23·21-s + 0.213·22-s − 0.983·23-s + 0.446·24-s + 1.67·25-s − 0.958·26-s + 0.508·27-s + 0.883·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 64T \) |
| 11 | \( 1 + 1.77e6T \) |
| good | 3 | \( 1 + 1.59e3T + 1.59e6T^{2} \) |
| 5 | \( 1 + 5.71e4T + 1.22e9T^{2} \) |
| 7 | \( 1 - 5.50e5T + 9.68e10T^{2} \) |
| 13 | \( 1 - 2.35e7T + 3.02e14T^{2} \) |
| 17 | \( 1 - 1.01e8T + 9.90e15T^{2} \) |
| 19 | \( 1 + 1.12e8T + 4.20e16T^{2} \) |
| 23 | \( 1 + 6.98e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + 5.69e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + 2.48e9T + 2.44e19T^{2} \) |
| 37 | \( 1 - 1.34e7T + 2.43e20T^{2} \) |
| 41 | \( 1 - 3.85e10T + 9.25e20T^{2} \) |
| 43 | \( 1 - 9.49e9T + 1.71e21T^{2} \) |
| 47 | \( 1 + 1.90e10T + 5.46e21T^{2} \) |
| 53 | \( 1 + 2.12e10T + 2.60e22T^{2} \) |
| 59 | \( 1 - 1.43e11T + 1.04e23T^{2} \) |
| 61 | \( 1 + 4.46e11T + 1.61e23T^{2} \) |
| 67 | \( 1 + 2.32e11T + 5.48e23T^{2} \) |
| 71 | \( 1 - 1.59e12T + 1.16e24T^{2} \) |
| 73 | \( 1 + 1.31e12T + 1.67e24T^{2} \) |
| 79 | \( 1 + 6.52e11T + 4.66e24T^{2} \) |
| 83 | \( 1 + 4.15e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + 3.61e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + 1.33e13T + 6.73e25T^{2} \) |
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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
−14.68450865446069579800254693425, −12.24105934152496135459761733453, −11.24607953926575255725377352193, −10.93230808130303376214251103328, −8.374812140758873910276229881054, −7.56321406342291859483157578567, −5.63741494300388003850907786446, −4.08304624777606676426899443657, −1.27910233354001605159055504260, 0,
1.27910233354001605159055504260, 4.08304624777606676426899443657, 5.63741494300388003850907786446, 7.56321406342291859483157578567, 8.374812140758873910276229881054, 10.93230808130303376214251103328, 11.24607953926575255725377352193, 12.24105934152496135459761733453, 14.68450865446069579800254693425
