| L(s) = 1 | − 1.21·2-s − 3-s − 0.512·4-s − 1.58·5-s + 1.21·6-s − 3.41·7-s + 3.06·8-s + 9-s + 1.93·10-s + 0.222·11-s + 0.512·12-s + 1.27·13-s + 4.16·14-s + 1.58·15-s − 2.71·16-s + 17-s − 1.21·18-s + 0.607·19-s + 0.813·20-s + 3.41·21-s − 0.270·22-s − 9.09·23-s − 3.06·24-s − 2.48·25-s − 1.56·26-s − 27-s + 1.75·28-s + ⋯ |
| L(s) = 1 | − 0.862·2-s − 0.577·3-s − 0.256·4-s − 0.709·5-s + 0.497·6-s − 1.29·7-s + 1.08·8-s + 0.333·9-s + 0.612·10-s + 0.0669·11-s + 0.148·12-s + 0.354·13-s + 1.11·14-s + 0.409·15-s − 0.677·16-s + 0.242·17-s − 0.287·18-s + 0.139·19-s + 0.181·20-s + 0.745·21-s − 0.0577·22-s − 1.89·23-s − 0.625·24-s − 0.496·25-s − 0.305·26-s − 0.192·27-s + 0.330·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 + 1.21T + 2T^{2} \) |
| 5 | \( 1 + 1.58T + 5T^{2} \) |
| 7 | \( 1 + 3.41T + 7T^{2} \) |
| 11 | \( 1 - 0.222T + 11T^{2} \) |
| 13 | \( 1 - 1.27T + 13T^{2} \) |
| 19 | \( 1 - 0.607T + 19T^{2} \) |
| 23 | \( 1 + 9.09T + 23T^{2} \) |
| 29 | \( 1 - 1.07T + 29T^{2} \) |
| 31 | \( 1 - 1.43T + 31T^{2} \) |
| 37 | \( 1 - 6.43T + 37T^{2} \) |
| 41 | \( 1 + 5.43T + 41T^{2} \) |
| 43 | \( 1 + 3.24T + 43T^{2} \) |
| 47 | \( 1 - 7.95T + 47T^{2} \) |
| 53 | \( 1 - 11.8T + 53T^{2} \) |
| 59 | \( 1 - 7.70T + 59T^{2} \) |
| 61 | \( 1 - 8.79T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 + 8.36T + 71T^{2} \) |
| 73 | \( 1 + 2.88T + 73T^{2} \) |
| 83 | \( 1 - 16.4T + 83T^{2} \) |
| 89 | \( 1 - 5.07T + 89T^{2} \) |
| 97 | \( 1 - 11.9T + 97T^{2} \) |
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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
−8.083940928737787768202246629763, −7.51982968565917361341910440540, −6.67652173476336914184010850772, −6.01738500434764084504622532287, −5.14258825492691647247882066377, −3.96971867908262462433415160141, −3.76029733103692430907866501703, −2.28461441575697874509141434725, −0.884948626000903430195523876598, 0,
0.884948626000903430195523876598, 2.28461441575697874509141434725, 3.76029733103692430907866501703, 3.96971867908262462433415160141, 5.14258825492691647247882066377, 6.01738500434764084504622532287, 6.67652173476336914184010850772, 7.51982968565917361341910440540, 8.083940928737787768202246629763
